There is currently no means for the Earl to disclaim his title, if it has been longer than the permitted period since he inherited it. He also could not do so if he was the first holder, since the 1963 Act only applies to inheritance, and he cannot disclaim a life peerage. The reasoning from 1963 is that the Act was made for people who wanted to sit in the House of Commons, but could not do so due to an accident of birth - something they had no say in. For a life peerage, or peerage of the first creation, they should have refused the honour at the time it was offered.
The situation is now a bit different thanks to the removal of all but 92 hereditary peers from the House of Lords in 1999, and the resignation provisions in the House of Lords Reform Act 2014. If the Earl were one of the 92, then he could not disclaim the peerage until he resigned, or was disqualified under other provisions of the 2014 Act. On resignation, he would become able to vote in general elections, stand as a candidate, and sit in the Commons if elected. (By the way, Church of England bishops who are members of the Lords can also resign, but do so in a different way.)
So any hereditary or life peer who feels that they are disenfranchised by virtue of sitting in the Lords also has the means to leave that chamber. That makes it difficult to make a human rights claim on the basis of Article 3 of the First Protocol. The European Court of Human Rights has upheld other kinds of voting restrictions in parallel circumstances, such as Ahmed and others v United Kingdom [1998]. In that case, several local government officials were barred from certain political activities, including standing for election, and brought a claim against the UK concerning violation of Articles 10 (free expression), 11 (free association) and 3 of Protocol 1 (free elections). The Court found that the restrictions were justified, and in particular that the applicants could avoid them by resigning.
An opposite case was X v United Kingdom [1978] in which the applicant claimed a violation of Article 6 (right to a fair trial) because he claimed to have the right to sit in the Lords but his petition to the Queen was denied. The Commission found that participating in the legislature is part of public law as opposed to a "civil right" within the scope of Article 6. Also, granting or withholding an honour has traditionally been held to be immune from judicial review - cited for example in Council of Civil Service Unions v Minister for the Civil Service [1985] AC 374 - although this conclusion may no longer be completely solid since the well-known 2019 Miller case. That would apply for aspects of a peerage which are purely ceremonial in nature.
The Earl might want to argue that his free expression is impaired by having a hereditary title at all. But there is nothing stopping him from calling himself by an ordinary name. British law doesn't require him to be called the Earl in any context, and there are plenty of peers who use their normal names while working at normal jobs. He has a defined place in the order of precedence, but I am not convinced that the situation would arise where he insisted on going to a fancy dinner and had a contretemps with the host over the seating arrangement that amounted to a human rights violation. Equally, he has the right to wear an Earl's coronet at a royal coronation, but our newly republican Earl would probably just not go.
In the case of the Earl Marshal - a hereditary office which goes along with being the Duke of Norfolk - there are non-trivial duties involved, and an Earl Marshal might justifiably want to stop doing them. The past procedure has been for the Crown to appoint a Deputy Earl Marshal instead, as was done in 2000 when the Duke at the time was unwell. It would seem that a similar manoeuvre would work for any other hereditary offices that were more than nominal. If the Crown refused to do this, then perhaps a claim could arise under Article 14 (discrimination on the grounds of birth), but it would require a particularly intransigent Crown.
Hereditary peers may hold property in a variety of entertaining feudally-derived ways, in addition to more familiar modes. The most relevant scenario to imagine is that our Earl owns Exampleshire Castle in "fee tail", meaning that he can't dispose of it even though it is horribly expensive for him to keep paying for repairs to its leaky roof - it must pass to his eldest son, and so forth. Even if the Earl is able to disclaim his peerage, he may still be on the hook for the property, depending on how everything is structured. However, there are generally legal workarounds these days, beyond the scope of this question, unlike when entails were so important to the plots of historical stories featuring the aristocracy.
There is a current campaign by daughters of some peers to change the inheritance laws away from male-preference primogeniture, including on human rights grounds. It remains to be seen how that will work out, but a key difference here is that they are losing out by not being able to inherit, while the Earl is able to divest himself of all the disadvantages of the title, by simply not using it.
