Middlesex head into the final day of the match requiring 209 runs to claim victory, after Burns' 121 and Harinath's 109 set Surrey on their way to 341 all out in their second innings.
Burns and Harinath's 217-run partnership represented a Surrey record against Middlesex for the second wicket, eclipsing the 184-run stand between Andy Sandham and Andy Ducat at the Oval in 1927. In reply, Middlesex reached stumps on 45 for one after Gareth Batty had Chris Rogers lbw for 19.
Worcestershire's lower-order batsmen made the most of a window in the weather late on the third day against relegation rivals Lancashire at New Road.
The match was at standstill for more than five hours before Gareth Andrew and Ben Scott came out to add 27 runs and lever their side up to 251 for seven before Daryl Mitchell's tactical declaration.
Lancashire reached 10 without loss from seven overs before the close, and where the game goes from here depends on whether the captains are prepared to strike a deal.
Nottinghamshire need to score more than 340 runs on the final day if they are to beat Durham and maintain the pressure on leaders Warwickshire.
Opener Mark Stoneman was the star for the visitors, hitting his first century of the season, as Durham closed their second innings on 301 for seven, 341 ahead, to raise hopes of a third straight win.
Stoneman made 114 from 147 balls, with 14 fours and a six, while skipper Paul Collingwood was unbeaten on 51 at the close.