[lgessler]

Hm, is this really a dangling modifier? Something's a dangling modifier when its modificand is unexpressed or ambiguous, but "for cancer research" has a clear modificand, the noun phrase headed by "minutes".

Rather, I think this is just some ambiguity about what the "few minutes" are for.

[stavros]

The joke is that Cancer Research is a UK charity. Sparing a few minutes for them means listening to their pitch, but the author takes it to mean as it being for the lowercase cancer research, hence a few minutes do nothing.

If you had understood the joke in the first place, please unread all the above.