After the return from Israel, Gingrich told the reporters that he felt ''snubbed'' by Clinton. he said that the snub contributed to the government shutdown in late 1995. It was one of the reasons that the Republicans lost in the 1996 election. So what would happen if Gingrich kept his mouth shut?
It wasn't one of the reasons the Republicans lost the '96 election.
However, if Gingrich can win the war over the budget shutdown (which, please note, him not being childish doesn't guarantee) then I'm willing to bet they could pick up some House and Senate seats and perhaps some Dole butterflies—maybe he stays in the Senate, maybe picks a more likely VP candidate than Kemp, and perhaps he generally gives a better showing for himself.
If so a '96 election where Clinton wins again but his downrace impact is worse than OTL is going to leave the Congressional Democrats mean and nasty towards him with possible future consequences (a stronger challenge to Gore in '00, for starters).
Furthermore a Clinton Presidency that leaves the non-Presidential level Democratic Party even worse off than OTL (in Congress, state Congresses, and Governors) will probably also have future consequences.