Orient's clash with Jose Mourinho's men was called off after a number of players at the Sky Bet League Two club tested positive for coronavirus.
The third-round game was due to take place on Tuesday night, and under the rules of the competition Spurs should be awarded a bye to the fourth round, where they would face Chelsea or Barnsley.
However Travis believes axing Orient from the competition – after they accepted Spurs' offer to pay for testing which is no longer mandatory – would cause other clubs to resist a similar opportunity.
Travis told BBC Radio 5 Live: "If I was doing this again I would not do the testing – this is an incentive not to test and that is bad for football and bad for health and safety.
"That's why we can't be punished over this, because people should be incentivised to take the test, especially when Premier League clubs with their resources give us the opportunity to have all our players tested.
"Flexibility is needed, cool heads are needed to sit around the table and find a solution but the answer is not to incentivise people not to test."
The League Two club were clearly frustrated by the nature of the postponement, which came after Orient revealed that a number of their players had tested positive on Monday.
A statement from Orient read: "Tonight's Carabao Cup match between Leyton Orient and Tottenham Hotspur is not taking place this evening as scheduled.
"Discussions are ongoing between the relevant stakeholders in regards to the implications of the fixture not taking place tonight and a further update will be provided in due course.
"Leyton Orient, the EFL and the relevant authorities are conducting a thorough review of the club's Covid-19 secure procedures with the view to reopening the stadium and training ground as soon as possible."
One thousand fans were also due to attend the fixture and Travis turned his fire on the Government, who have reversed plans to allow supporters back into elite sports venues.
Travis added: "We're in a crisis where football is not getting supported by the Government.
"If I had to say who are the real villains here, it's the Government. Every week they chop and change their minds about what's happening with football.
"They bring in a few weeks ago a sensible programme of having 1,000 fans within the stadium...as far as I know that has been successful [and] I think that could have continued, but we've got that knee-jerk reaction again today.
"What the Government needs to recognise is if they want live sport they're going to have to support it and other sports financially."