Yep. Some attention should be paid, for a few reasons.
One, to get the aero benefit of fancy aero wheels, you need to width-match them correctly to their tires. Ideally the tire and rim together form one nice aerodynamic shape.
Two, the tires can mount a bit weird if the differences are really extreme. In particular, people mounting ultra-wide tires on ultra-narrow rims sometimes report that the handling is squirmy (presumably because the tire's sidewalls form a too-narrow flexible "neck" where they contact the rim sidewalls).
Most of the time, within reason, if your bike has the clearance for it, things work out just fine. Right now my hardtail actually has a hybrid tire on the back, and it's still basically behaving in a tire-like manner.
If the change isn't really big, you won't even need new tubes. Tubes are able to stretch to enormous sizes without failing, and they're usually rated for a range of tire widths. Heck, they're sometimes even rated for a range of wheel diameters; a lot of tubes are spec'd for both 700c and 27" wheels.
That said, if you're going
down in size, you'll probably want a new tube anyway. Installing a tire with a too-large tube can be difficult; it's more challenging to pop the tube into the rim channel, and you have to be extra-careful to avoid pinching it between the tire's bead and rim sidewall.
If you're doing a small or moderate width
increase, you can probably happily just use the tubes you have.
They think it will be delivered today so hopefully I can get it this weekend.