Hereford ended a run of three successive league defeats as they battled to gain a point at Cheltenham.
The hosts came closest to scoring in the first half but Darryl Duffy's effort from Marlon Pack's corner was cleared off the line by Stefan Stam.
Hereford's best chance fell to Sam Winnall, who beat two home defenders but fired an angled shot well wide.
Bulls goalkeeper David Cornell ensured it finished goalless with an acrobatic fingertip save from Pack's shot.
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Cheltenham Town manager Mark Yates told BBC Gloucestershire:
"It was a disappointing scoreline but the performance was excellent. We were the team that was going to win it but we didn't make the keeper work enough.
"Sometimes it just doesn't go for you, we had a couple kicked off the line, a couple of bad decisions when we've been in good positions and Josh (Lowe) missed a chance that I've never seen him miss in my life.
"We had 24 goal opportunities so we've got keep working at being more clinical rather than being ruthless and just getting the shot away."