Basically there are two options for residential wiring (keep these gauge/breaker sizes together):
14 gauge wiring with 15 amp circuit breakers, or;
12 gauge wiring with 20 amp circuit breakers.
Either can safely be used with 15 amp receptacles. 15 amp receptacles are cheaper than 20 amp and rarely do applicances need more, so it is common to see 15 amp receptacles installed on 12 gauge/20 amp circuits. OTOH I had to swap out a 15 amp receptacle for a 20 at the washing machine because it had a 20 amp plug, but our entire house is 12 gauge/20 amp circuits (or bigger), so it was OK.
Swapping out a 15 amp breaker for a 20 on a circuit with 14 gauge wiring is even a more siginificant code violation and potentially much more dangerous than installing a 20 amp receptacle on a 14 gauge/15 amp circuit. Now, if you never run more than a 15 amp load through that circuit or receptacle, in theory you're safe, but the potential for overload would exist and that's the risk here.
These are all time tested rules that are there for protection of the public. If you read the NEC (National Electric Code, that virtually all U.S. building codes are based upon) you'll find the residential to be much simplier but "stiffer" (prescriptive) than the general electrical code where there is somewhat less chance of "amateur playing around".
Insurance companies could refuse a claim for any work not done to code (which would require a properly licensed electrican doing the work under permit and obtaining final approval by the code authority having jurisdiction).