A leather drum is an atonal musical instrument, so it cannot be used to play the passtune. It is one of the two drums. When played, you will "beat a deafening row!" Applying a leather drum will awaken (and possibly scare) nearby monsters – including pets – and abuse your wisdom.
The leather drum is not especially useful for most players, who will probably discard it or stash it for later polypiling, hoping for a more valuable tool. One role that may find it worthwhile to carry is the Knight, as it can be used to awaken sleeping monsters and avoid an alignment penalty (it's still not the best tool for this job, as lighter and/or tonal instruments are also available). Since using it abuses wisdom, you should follow beating the drum with some activity that exercises wisdom, such as engraving Elbereth, or restores stats lost to abuse, such as applying a unicorn horn.
Many travelers have seen the drums of the great apes, and
some have heard the sounds of their beating and the noise of
the wild, weird revelry of these first lords of the jungle,
but Tarzan, Lord Greystoke, is, doubtless, the only human
being who ever joined in the fierce, mad, intoxicating revel
of the Dum-Dum.