I'd say Mario Kart would be more likely to ruin relationships, haha. Mario Party to a certain extent, but I've had intense moments in MK with my friends and cousins than MP.
Mario Party and Mario Kart are probably the only games I take relatively seriously to the point where I will trash talk the crap out of anyone, does not matter what their relation to me. If I recorded everything my boyfriend and I say to each other while playing MK, it would probably sound like domestic abuse. Apparently his roommate finds out intense gaming and screaming amusing though.
Otherwise, co-operative/multiplayer games are played out of fun. After experiencing the LoL community, I'd much rather avoid them altogether and has been one of the main reasons why I stick with single players.