If the Celtics make the finals it will mean one thing has changed, and that is this team can beat the best team in the East. Conventional wisdom is that the Cavs and LeBron still have at least a couple years of dominance left, maybe more while LeBron is still in his prime. Beating them would change that thinking.
Secondly, how they perform against the Warriors would influence them. If they've shown they can beat a healthy Cavs team, the smart move would be to make moves that enhance their ability to match up with the Warriors. The Cavs are who they are, their ability to drastically improve their team the next couple years is limited.
Having said that, I don't think it changes Ainge's thinking that much. When they signed Horford, they were clearly intending on making a run within that 4-year window. Also, they have cap space that will be eaten up by Thomas and/or Bradley next season so they were always going to have to strike this offseason. Otherwise, they lose that space for nothing. If Hayward can't be convinced to join, I can see them being more open to dealing the BKN pick (and perhaps Jaylen Brown) if it's not #1 overall for a star like Paul George, but of course that depends on him actually being available.
We're not going to beat the Cavs but it's an interesting question nonetheless. Still, I don't think it means Ainge makes a drastic move (something like overpaying in a deal for a star) that he wouldn't make otherwise.
Probably the only time Ainge made a risky "win-now" trade was the fifth overall pick for Ray Allen. Yes, it ended up being worth it but only because Green ended up disappointing and they also got Garnett. It's not clear whether Ainge knew Green was going to be bad because he dealt for the guy a few years later. It's also not clear whether Ainge had a wink-wink deal with McHale and/or Garnett when he dealt for Allen. The trade for Allen doesn't make as much sense without Garnett coming here.