You won't be lost or anything if you aren't familiar with Chekhov, but some of the jokes/references might pass you by. The setting, the plot, the style of dialogue in spots are all allusions to Chekhov so it's much funnier if you know what Durang's spoofing.
Did you have to know every flop referenced in [tos]? No, but it was funnier if you did.
Really, if you know The Cherry Orchard, Three Sisters, Uncle Vanya, The Seagull and Ivanov you'll be set.
Marie: Don't be in such a hurry about that pretty little chippy in Frisco.
Tony: Eh, she's a no chip!