Fans face a overnight stay or 500-mile round trip for the third round clash after it was moved to 7.55pm on January 8.
The last train to leave the Devon city for Liverpool departs at 6.25pm.
Reds fans’ group Spirit of Shankly, who are long-time campaigners for supporters’ rights, have attacked the plans saying it does not take into account the safety of thousands of people travelling or the general inconvenience it will cause.
“It shows the contempt and lack of regard with which they hold football supporters,” spokesman Jay McKenna told Press Association Sport.
“It is a physical manifestation of their greed.
“For the BBC to schedule a match for a Friday night when the only way to get there is by car or coach [shows] they don’t care whether supporters can get there and back safely at a decent hour.
“We are relatively powerless but I think this will have a negative impact on a lot of Liverpool supporters who will not go – and that is not about the travel, it is the fact they will have to take Friday off work – and those that work Saturdays will have to take two days off.
“I will probably be watching at home and some of our committee members have already said they can’t justify taking two days off.”
McKenna also criticised the authorities and broadcaster for failing to consult adequately over their plans for the match, although he accepts fans’ views are rarely taken into consideration.
“We are not the first club to suffer from this – and we won’t be the last – as FC United had their match moved to a Monday but what angers us is they don’t actually speak to supporters to ask what impact it will have,” he added.
“Both clubs but mainly the FA and BBC should be saying, ‘We realise it is going to impact on you and this is what we will do to help.
“The FA and BBC should be offering financial compensation with subsidised tickets and travel.”