Maybe a gittip feature (I don't know if it already does this) is an option to make all donations anonymous so that the recipient does not know the source. This should remove the conflict of interest then.
As whit537 mentions in a sibling comment, all Gittip donations are already anonymous, but blog posts like John's here end up removing the anonymity for particular donations.
That doesn't stop bribes. I could "gittip" you a lot of money. It comes up as anonymous, but I tell you that it was me.