You could say the joke "bombed" or "fell flat" (which has already been added as an answer). However, this is more in relation to how the joke is received, since the joke could indeed be funny on its own, but it "bombed" with the crowd it was delivered to. Or even the delivery itself of the joke "bombed". Another provided answer that falls into this category is: "flopped". From an article which uses a few of those terms: "Bombing (in Comedy)".
Definition: To bomb as a comedian means that your jokes don't go over with the crowd -- i.e., they get no laughs.
Other forms of expressing this usually refer to death in one way or another, for example I have heard some use the expression: "the joke died on the operating room table".
If what you are trying to communicate is that the joke was truly unfunny, and not just its delivery or reception, then you can say (as mentioned in other answers) that the joke was "lame" (where "lame" literally means "cannot walk properly", or in reference to a limb being disabled), or any other such qualifier (eg: "bad", "dumb", etc).
Yet a third approach, sometimes when a joke is just terrible, some people (for example I have a friend who does this all the time) just respond by saying "womp womp" which is meant to reflect the following sound made by a trumpet, tuba, trombone or other such brass instrument:
Which is often a shortened version of this:
womp womp extended on trombone