The arrests came during a busy opening to the Isle of Man TT for the Isle of Man Constabulary, which has also dealt with offences including suspected drug-driving and drunkenness.
A police spokesperson said: ‘The public is reminded that you must not enter closed roads at any time.
‘Riding, driving or even stepping onto a closed road places you and others at risk and you may be placed before the next available court.’
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Police have not released further details about the two men arrested in relation to the road closure breaches.
The warning comes after a serious and complex incident in Ramsey on Monday left eight spectators and a rider injured during TT practice sessions.
As previously reported, a rider was also arrested on Monday on suspicion of speeding and drug-driving.
The rider was travelling from Kirk Michael to Ballig and was allegedly recorded travelling at more than 118mph in 50mph zones.
Officers said the rider was found to be under the influence of drugs after being stopped.
Two other individuals were arrested on Monday for drunken offences, while several more people were held in police cells overnight for a range of offences.
Police have also urged members of the public to respect TT marshals throughout the festival period.
The spokesperson said: ‘Marshals are a pivotal part of the TT and give up their time to help keep everyone safe and none of this would be possible without them.
‘Listen to their instructions and follow their safety advice. Your cooperation helps protect competitors, spectators and the wider event.’