Most cartridge-based games will not have their main chips die anytime soon. The save mechanisms are different, but for the game chip itsself it's not a problem... except for:
A small amount of old cartridge games used EPROMs instead of masked ROMS, and those can die. I used to open a lot of 2600 cartridges and you do see the occasional EPROM.
Newer homebrew/indie releases can have the same problem. Battlesphere for the Jaguar is on EPROM. If you aren't doing a huge run, making masked ROMS isn't feasible.
Note that this has nothing to do with game saves on NES/SMS cartridges and the like, that's another whole issue.
This is important because EPROMs only have a 25-year lifespan, compared to the indefinite lifespan of a normal masked ROM. But since all officially-released cartridge games I know of since the NES generation use masked ROMs and not EPROM, it's not a big problem except for 2600 fans.
The second issue is the contacts. As has been said, while blowing on carts makes them work (and I end up doing it far too often out of impatience to just get the game working), it's not good for the carts since it degrades the contacts. Over time this will cause problems... use the Q-tip and isopropyl alcohol solution or buy a generic cart cleaning kit. It's well worth it.
Sorry, but most cartridge based systems have copper connectors, which, over time, will degrade.
The move you use them, the worse it will get.
I've heard that a cart game should last about 25 years or so.
In the case of TG16/PCE, where they used gold connectors, your mileage may varry.
So, youre considering pending a bucket of cash on Neo Geo home carts, it wouldnt be a wise investment.
Not that I advocate the open distribution of illegally dumped roms, but thank god for their existance.
If they've been taken care of well they'll last a lot longer than that... and anyway, Neo Geo had new games in production until three years ago. If quite a few 20 year old NES carts are still working, why shouldn't many Neo ones for decades to come? Particularly if you don't blow on the contacts...
Saving is a different, and much more complex, issue. Some people have mentioned the fact that most cart games that save use batteries, but unfortunately it's not anywhere near that simple... I'll have to go by platform.
First, as far as I know no pre-NES console had in-cart saving (though I could be wrong about that). Lynx has no saving either. Turbografx-16 iwill be addressed later (it's unique).
Second, note that numerous games for every platform that has saving don't support saving; the percent of games with saving is very low in older platforms and gets higher with time; the GBA probably has the highest percentage of games with internal saving of any cartridge-based system ever, but the NES has very few that are affected.
For the NES, Super Nintendo, Game Boy (original), Game Gear, Master System, and some other platforms, all games that save have an internal save battery (this system is called SRAM -- battery-backed RAM that requires power to keep data on it). To change it you'll need the special screwdriver that opens the cart, a replacement battery (CR2032 for the major console systems, smaller ones for handheld games), and a way of getting off the old battery and attaching the new one, preferably a soldering iron (and solder)
Some cart games, however, do not have batteries. That is, they use forms of flash memory -- EEPROMs, Ferroelectric RAM (FeRAM), or Flash RAM. For instance, all Neo-Geo Pocket/Pocket Color and Nintendo DS games are on flash cards, so no games on those platforms have internal batteries. Lynx is the same way apparrently.
Atari Jaguar: all games that save use EEPROMs, apparently. Jaguar CD: unknown?
Game Boy Color: most games, and all black carts (dual-mode), use batteries. A few GBC games use EEPROM; you can tell since the carts are transparent, the game has saving, and there's no battery in it.
Game Boy Advance: all four save systems (Flash RAM, FeRAM, EEPROM, and SRAM) are used. Most SRAM games are older titles; most FeRAM games are newer. I don't know of any list that can tell them all apart -- some emulation sites have lists, but they can't determine between FeRAM and SRAM, so the lists are of limited use (if you're trying to determine which of your games have batteries in them without opening the carts).
Genesis: almost all games use batteries. Three games have FeRAM (Sonic 3, Mega Man: The Wily Wars, and one other). Three others have Flash RAM (I think it's not EEPROM anyway): NBA Jam (not TE) and two others. Genesis FeRAM is said to have a nearly 100-year lifespan, but I'm not sure how true that is, and that requires the cart to keep working that long, which is of course far from being a sure thing... (never heard any lifespan times for EEPROM or Flash RAM, just that they eventually can't be saved to once flashed enough times and then can only be read)
Nintendo 64: like the GBA, the N64 uses several systems -- EEPROM, SRAM (battery), and Flash RAM, as well as no saving and Controller Pak (memory card) saves. Half of all N64 games require controller paks to save, including most third party titles; all first-party games use internal save. Of the US-released titles, most N64 games with internal save use EEPROMs. Twelve games use SRAM and a dozen more Flash RAM. The SRAM games include Zelda: Ocarina of Time (but not MM, that's flash), Ogre Battle 64: Person of Lordly Calibur (note that this game also supports controller pak save), 1080 Snowboarding, F-Zero X, Resident Evil 2, The New Tetris, Super Smah Bros., and a few others. It's unfortunate that the SRAM games include so many of the great games on the system, but oh well... at least those batteries can be replaced when they die, unlike dead EEPROM chips, though those of course have far longer lifespans, mostly negating the point...
TurboGrafx 16, Sega CD, and Sega Dreamcast: an internal (not easily replaced) rechargable battery hold the internal save files (settings only for the DC). 3D0 and CD-i too I think. There are a couple of TG-16 games that have interal save batteries, but the vast majority save to the system (note that the TurboExpress portable has no save battery, so it cannot save games).
Saturn: user-replacable battery (CR2032) holds the internal save.
Neo-Geo and Nintendo 64: memory cards (Controller Paks for the N64) use SRAM (batteries are in the cards). Well hidden, in the case of Neo-Geo... (though you could use a new SRAM PCMCIA card, I believe, if your old card dies; for the N64 the best option is copying the files to a PC with a Dexdrive or something like that and replacing the battery...) [Neo-Geo cartridges have no save systems within the carts. Card only.]
Playstation 1/2, Dreamcast, Saturn, Gamecube, Xbox, Xbox 360: proprietary flash memory cards hold save files (for the Saturn this refers to the memory carts; those are flash.).
Xbox, Xbox 360, Playstation 3: internal hard drive save option.
Wii: internal flash memory saves your files (not just config settings like on some previous systems I think).
Wii, PS3: support common format flash cards for media (SD, etc).
Come on guys. Cartridges may not last forever, but everyone of mine still work. I even have a couple Fairchild carts, and they work just fine. If you dont know what that is, look it up, to find out how old it is. Here is a hint, it is older than the Atari 2600!
All of my cart games work... excepting three Genesis games, which I got used and never worked. One doesn't respond at all (the Sega CD boots); one just goes to a permanant black screen; and one displays the Sega logo, and then stops there... I've tried cleaning them as well as I could, but no change.
As for saving, I only have SNES/Genesis/N64/GB/GBC/GBA (and Gamecube), but while all of my N64, GBA, and Genesis games with batteries are fine so far, some of my GB, GBC, and SNES battery-backed games have dead or dying batteries in them. It's when I realized how annoying an issue this would be that I started to look into it (as this post probably shows)...