The corporation, which recently lost the rights to broadcast the tournament to Sky, is beginning its coverage at 1.45pm after its regular daytime shows including Bargain Hunt.
Writing on Twitter, Geoff Critchley said: “Heir Hunters and Bargain Hunt. More important to the BBC than The Open. Thank goodness Sky will be taking over.”
Another viewer, Kelvin Gomez, said: “This is the reason why Sky will have the rights to the Open next year - a sad finale for the BBC.”
The tournament, at St Andrews, Fife, was extended into today after bad weather stopped play over the weekend and the Open’s leaders will not start play until after the BBC coverage begins.
A BBC spokeswoman said: “The BBC has brought unprecedented coverage of this year’s Open to the widest possible, free-to-air audience - with extended scheduling across all days of competition, including two extra hours on both Friday and Saturday.
“Our key aim today is to fully tell the most important story - who will win the Open - and make the very best use of our resources on this extra day’s play.
“We will be following the leaderboard top 10 from the start of their rounds to their finish, plus the continued action from those that started before them - so audiences won’t miss any of the key action as it unfolds.”
More than four million people tuned in to watch the tournament on BBC One yesterday.