Issues, derail and or extra wear and tear on the locomotive. Or they simply won't go as they'll get stuck, and you could burn out motors or drive boards as it's trying to push the train around. another issue would be with pickup rollers. If you go on tighter curve track than it's speced for, the rollers could fall off of the center rail. This can cause power shorts, derails, or damage to the pickup rollers.
Now, having said that, I run my trains under the speced min curve all the time. One of the 1st things I do with a new rolling stock is find out just how tight it can go. I go slow and watch it to make sure I don't damage it. Most freight and passenger cars can do O36, unless they are coupled to another car. Then some of them long ones will hit, or clothesline on a curve. Most locomotives can go down one sized curve, sometimes more like John pointed out, you just can't pull anything behind it. On steamers sometimes the locomotive can go tight, but the tender can't. Here's a list of some of my locomotives:
K-Line Big Boy - spec O31, does O27 just fine.
Lionel Visionline Big Boy - spec O72, locomotive does O54, tender will do O63, but doesn't really like it.
Lionel Heavy Mikado - spec depends on where you read it as it's been listed as both O36 and O54, run mine on O36 all day long without issues
MTH Protosound equipped PRR T-1 Duplex - spec O72, just barley got it to do O48 but the motor was trying too hard (Pulling too much current.), so O54.
Just keep track geometry in mind. Like my Mikado. Sure it can do O36 curves, and switches, but I have to be really careful on O36 S-curves. When backing the tender has come off the track. The VL BB, sure it could do O63 curve and switch, but any S-curve smaller than O72 will stop you in your tracks (Yup, I went there with a pun.). So best thing is if you really need to go smaller, lay some track out and slowly and carefully push/run it through the track and see how comfortable you are with it. If you have a LHS that you are a regular at, they may even help you try it in the store before you buy (Did that with Lionel and MTH auto racks on O31. They do it, but not coupled, and will hit switch stands.).