Although pleased with his team's performance against opponents tipped for promotion, the manager was frustrated not to take maximum points from the game.
Ross Hannah's exceptional strike gave the Blues an eighth-minute lead and the 29-year-old spurned a glorious chance for a second before half-time.
Cheltenham competed well but found it difficult to prise open the home defence until three minutes into the second period when Jon Worsnop misjudged his punch and Kyle Storer hooked home the equaliser.
Robins keeper Dillon Phillips denied Hannah a second with a breathtaking save and Gary Johnson's men struck the crossbar through Jack Barthram, and in the end both managers had to be content with a share of the spoils.
Burr said: "It was a high tempo game, end-to-end with a lot of gaps opening up later on in the game but we're very disappointed in our dressing room because the lads have worked hard again as you would expect but the goal that we've conceded was disappointing from our point of view.
"In the first-half I thought some of our passages of play were very good and these won't be far away come the end of the season so it was a good benchmark for our lads. We've got some exciting players there who are exciting the fans here but we're just a little bit disappointed.
"Ross was a little bit disappointed he didn't get the second to give us a bit of a cushion and in the second-half their keeper has made an unbelievable save to keep them in it.
"We're just a little bit down because these lads expect to win, which is a good thing."
LOOK: Action from the Blues draw with the Robins on Tuesday night
The draw extended Chester's unbeaten start to four games and Burr accepted his disappointment at not beating a team that were in the Football League until last season was a positive.
"I said that to them and it just shows how far we've come and how we're developing as a squad as well," said the 55-year-old.
"It's still early days with the squad but I've been delighted with how they're working for one another and the team spirit about the place is fantastic.
"With the start we got off to and if Ross puts it a fraction either side, and that's what football's all about sometimes fractions.
"They had a few chances with one that's hit the bar and probably had their best chance in the first-half when it was cut back by Amari Morgan-Smith and I think Danny Wright hit it over but I can't really think of too much after that apart from the header Jon Worsnop saved
"Little things tonight probably didn't drop for us in the box that we would have like but our movement from set plays was very, very good tonight."
Chester have a testing trip to fourth-placed Woking on Saturday (3pm). The Cards lost at Tranmere Rovers on the opening afternoon but have since beaten Bromley, Altrincham and Torquay United.