Varys & Petyr's Only Team Up

I wouldn't say Varys literally 'teamed up' with Littlefinger on that occasion, but rather simply didn't interfere.

And how could he interfere, even if he wanted to? Seemingly, the story about that tourney bet was Littlefinger's improvisation on the spot, Varys wasn't ready for it. He had already admitted not knowing anything about the dagger, and he couldn't just accuse Baelish of lying without providing an alternative explanation of the dagger's origins. And in any case, calling Petyr Baelish a liar would be a dangerous move, too dangerous for Varys, who must remain sly and obedient in order to survive the politics of King's Landing. Interestingly, perhaps he did try to hint Ned about LF, during the Hand's tourney, with a story of "a boy from the Vale, who owed everything to Jon Arryn", which Ned immediately took to mean the late ser Hugh, and Varys seemingly confirmed it. Maybe, Varys was just being very careful and didn't dare to speak openly, so he tried to give Ned a hint, and decided not to press the issue when Ned failed to understand.

Anyway, while there are great points in previous answers, and there's a possibility Varys and Littlefinger worked together on one or more occasions, personally I don't quite believe it, mostly because of "gods only know what game Littlefinger is playing", and "the Wolf and the Lion will be at each other's throats thanks to Littlefinger's meddling" (Varys to Illyrio, overheard by Arya).

Unless, of course, Varys was lying to Illyrio. At the moment, this is too much for me to process, but sometimes it does seem to me we should prepare ourselves to the revelations about Varys and Illyrio not exactly sharing goals. For now, though, I really believe they work together and Varys has no reason to lie to Illyrio regarding LF.