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Olise scored a penalty in stoppage-time to make it 2-2.
Leon Bailey's deflected strike after a surging run in the 74th minute sealed Villa's club record-equalling 14th successive home league win as they moved above City into third place.
Ten-man Chelsea beat Brighton & Hove Albion 3-2 at Stamford Bridge while fourth-placed Aston Villa needed a late Ollie Watkins goal to scrape a 2-2 draw at Bournemouth.
Trent Alexander-Arnold drilled in an 80th-minute equaliser with his right foot to deny City a 24th successive home win in all competitions.
Cole Palmer scored a late penalty to make it 4-4 at Stamford Bridge in London.
The result left Spurs second in the standings behind Manchester City.
City thoroughly dominated shell-shocked United in the second half.
Manchester United tasted a 1-3 defeat at Old Trafford while City won by the same margin against the Hammers.
In all, referee Anthony Taylor doled out 11 yellow cards along with Rodri's red, with seven of the yellows coming in a bad-tempered second half as Forest tried to leverage their extra man, but to no avail.
City started their title defence by coming from behind to beat Red Star Belgrade 3-1 at home.
Results 51-60 of 302
