Yes! This is an often more assumed problem than a real one for most travelers. Travelling to Iran will not be a problem.
Travel to Iran as an unmarried couple
Most hotels, like everyone else, won’t ask or care about your relationship status. In the case that they do, it is up to individual travelers to decide how they respond but no proof of marriage is required unless you are Iranian.
Although Iranian law prohibits unmarried couples from living together, which states that people of opposite sex cannot be alone together unless they're married or are family.
But in practical applications most Iran hotels don't enforce non-Muslim and non-Iranian the married couples rule. Luxury hotels, which often cater to foreigners, are especially relaxed, so chances are you won't have trouble booking a room.
But remember unmarried Persian couples are not able to get a hotel room. If you're married to an Iranian we advise you to carry your marriage certificate with passport as proof of marriage.
In the very worst scenario, you may be required to take separate rooms but this is almost unheard of, particularly for foreign travelers.
Holding hands is fine for married couples, but kissing or hugging and of course dancing in public are not acceptable.
Keep in mind that if you're stopped by the police because you're holding hands of your Iranian partner and you're not married, you might run into trouble.