Catholics are currently barred from the throne
A backbench bill proposed in Parliament by Hull MP Kevin McNamara would also end the bar on Roman Catholics succeeding to the British throne.
Mr McNamara's bill would also mean MPs would no longer have to take an oath of allegiance to the crown before they take their seats in Parliament.
Oath complaints
Some MPs who want a referendum on the future of the monarchy, such as Labour's Tony Banks, have crossed their fingers as they took the oath in a show of protest.
Mr McNamara told the Commons: "I am looking, in this bill, to strike at discrimination and intolerance in our society. I am looking to assist the process of inclusion."
He quoted the European Convention on Human Rights, which includes the right to free expression, as he voiced his opposition to the 1948 Treason Felony Act.
That law means anyone convicted of "imagining" or expressing their intention to abolish the monarchy - even by peaceful, democratic means - can be punished by life imprisonment.
Mr McNamara also branded the 1701 Act of Settlement as "extremely offensive" because it means Catholics neither be the monarch or marry into the royal family.
This week's furore over whether to allow Sinn Fein MPs offices in the House of Commons has reignited the arguments over MPs' oaths of allegiance.
The oath is one reason why Sinn Fein politicians will not take their seats in the Commons, although they also say they would not sit in what they regard as a "foreign parliament".
'Ending absurdity'
Even some MPs opposed to the decision to give Irish republican MPs offices, such as former Conservative minister Douglas Hogg, have suggested it is time to change the oath.
Calling for a "modern affirmation", Mr McNamara said: "I think we can get rid of the indignity and absurdity that we have had in the past from members taking the oath.
"We can return to a situation where it is treated with solemnity because people will return to a position where they are doing something voluntarily."
The MP's two previous moves to change the oath have fallen at the first hurdle in the Commons.
This time his bill did receive its formal first reading but not without meeting strong opposition from Mr Paisley.
Constitution's history
The DUP leader accused Mr McNamara of trying to "underwrite the constitution" and said the issues were too weighty to be tackled by a backbench bill.
Several other European countries, such as Sweden and Spain, kept the constitutional link between religion and monarchy, argued Mr Paisley.
"To say now that our hands should be put to the job of dismantling it is a strange thing because we don't know what they want in its place."
Despite passing its first vote, there is unlikely to be enough parliamentary time for the bill to become law.