Referee Neale Barry had to endure the wrath of the visiting fans after dismissing Lee for a second yellow card and Murray for a crude challenge in the dying minutes.
Despite having the advantage of numbers, especially following Lee's sending off, Bradford failed to find the killer touch and had to settle for a fourth consecutive home draw.
City made a bright start with on loan Sunderland striker Michael Proctor again impressing with his pace and touch but there was little goalmouth action to warm the crowd.
Derby's five-man midfield did well to stifle Bradford but they struggled to break down a well-drilled City defence with Robert Molenaar in command at the back.
It was almost 20 minutes before the Dutch centre half made a mistake and the Rams failed to take advantage of his wayward pass.
Derby midfielder Adam Bolder slipped the ball through to striker Izale McLeod but he could only fire wide from 18 yards.
After 25 minutes Derby's best move of the half created an opening for Murray but, after Malcolm Christie's clever step over, City keeper Steve Banks smothered Murray's low shot.
Molenaar made a vital challenge on McLeod as he threatened to skip past him in the Bradford box and it was the home side who finished the half stronger.
City skipper Ashley Ward had their best chance six minutes before the break when Proctor's pace stretched the Rams defence.
His low cross into the six-yard box should have been forced home but the ball flew up off Ward's leg and Derby keeper Mart Poom was able to grab it at the second attempt.
With Welshman Paul Evans seeing more of the ball in the Bradford midfield Derby had to survive another couple of scares before the interval.
A minute after the restart Lee got the first of his two yellow cards and it proved costly when he was given his marching orders after 71 minutes for upending Stephen Warnock in full flight.
Referee Barry was a busy man as the challenges came in thick and fast and there were few clear-cut openings for either side.
Evans almost broke the deadlock when he was put clear by Ward but his low drive was well covered by Poom.
City substitute Juanjo fired a good chance wide of the mark just four minutes from time but depleted Derby hung on to secure a third successive clean sheet away from home after Murray was given a straight red for his rash tackle on Warnock