Can you run a 15 amp plug off a 30 amp breaker?

No. This is one of the cleaner no answers in the code. Table 210.21(B)(3) lists the permitted receptacle ratings by circuit rating. On a 15 amp circuit you may use a 15 amp receptacle. On a 20 amp circuit you may use a 15 or 20 amp receptacle. On a 30 amp circuit the only permitted receptacle rating is 30 amps. There is no allowance for a smaller receptacle on a 30 amp circuit.

Why the 20 amp circuit exception does not extend

People reason from the familiar case of a 15 amp receptacle on a 20 amp circuit and assume the pattern continues. It does not. A duplex 15 amp receptacle is rated for 20 amps of feed through, so multiple 15 amp devices can safely share a 20 amp circuit under 210.21(B)(3). A 15 amp receptacle has no such rating relationship to 30 amps. If a cord and plug load faulted or a device drew 25 amps, the receptacle contacts and the device cord would carry far more than their rating and the 30 amp breaker would never see a problem.

Section 210.21(B)(1) covers the single receptacle case. Where a single receptacle is installed on an individual branch circuit, it shall have an ampere rating not less than that of the branch circuit. So a single 15 amp receptacle alone on a 30 amp circuit is still prohibited.

What to do instead

If the load is a 15 amp cord and plug appliance, the correct move is a 20 amp or 15 amp branch circuit. If the 30 amp circuit already exists in 10 AWG and you want general receptacles at that location, change the breaker to 20 amps and install 20 amp receptacles. A 10 AWG conductor on a 20 amp breaker is perfectly legal, since 240.4 sets the maximum protection, not a minimum. The only nuisance is landing 10 AWG on a standard receptacle, which usually means a short 12 AWG pigtail inside the box under 110.14(A).

If you need both a 30 amp load and normal receptacles, run a second circuit. Do not install a 30 amp to 15 amp adapter as a permanent solution, and do not swap the breaker down without confirming what else is on the circuit.

Related sizing

Section 210.24 summarizes branch circuit requirements. A 30 amp circuit requires 10 AWG copper minimum, and 240.4(D)(7) caps 10 AWG copper overcurrent protection at 30 amps. Section 210.23(B) limits cord and plug connected utilization equipment on a 30 amp circuit to 24 amps, that is 80 percent of the circuit rating.

Field takeaway

Thirty amp circuits get 30 amp receptacles. To serve 15 amp plugs, change the breaker to 20 amps and use 20 amp devices, or pull a separate circuit.