User-side locks, including Mutexes do not block interrupts. This is important, since often a Mutex may be used to protect a read from a file on disk, or protect the reception of packets from a network, and such, where the result is reliant on an interrupt.
In essence, "nothing" happens if there is an interrupt. Interrupts are handled by the OS, just like any other time. There is nothing your program needs to do in this case, it just takes a few microseconds or milliseconds longer to do whatever you programmed your task to do if this happens.
Some kernel-side locks, such as SpinLocks do indeed block interrupts (on that processor core) to ensure that other processes/threads do not get scheduled during this process. There are also restrictions on what functions the kernel can use under these circumstances - for example, it wouldn't be possible to call a blocking function (such as a sleep, wait-for-event or file-read or file-write) during this time, because that would, potentially, cause the kernel to lock up.