Does a contract matter
If I am understanding your situation correctly, when you were hired you had a certain understanding with your department and university about your duties, which included both teaching and research. Since you don't say otherwise, it sounds like there are "standard" such duties in your department. Also there was a standard contract, and by some strange mistake you were given and signed a different, nonstandard contract. After four years of fulfilling your standard duties, the nonstandard contract has been brought to your attention. Do I have all that right?
Now you are asking whether you should "put up a fight" about the fixing the mistaken contract. If your question is a legal one, of course get a lawyer who is an expert in the specifics of your situation.
But are you really contemplating fighting against what you verbally agreed to when you took the job and the job you have been actually doing for the last four years because a contract you signed says otherwise and the difference may work out in your favor? To me that sounds like horrible behavior. If your bank had noticed after four years that they had mistakenly added a zero to your account balance, would you also fight the correction of the error? Do you not have any ethical qualms about this?
As a justification you say that you wonder that if the situation were essentially reversed, correcting the oversight would be "a real fight with HR". First of all: is this guess grounded in any kind of history with the HR department at your university? Have they in fact been difficult with you in the past? If not it seems like you are simply assuming bad faith. Here's what I think would happen if you discovered your contract was all teaching and no research and tried to change it: there would be plenty of red tape, the entire process would take an order of magnitude longer than you felt that it reasonably should and would occupy too much of your time, but the final outcome would not really be in doubt due to the essential goodwill of the various parties, including people like your department chair/head who would come out on your behalf.
Aside from being ethically highly suspect, a decision to fight the change of your contract is likely to earn you the ill will of HR and other administration in your university and -- probably more importantly -- of the department chair/head and other faculty members. You are contemplating not doing your share of the departmental work because of some technicality that you wonder whether you might be able to get away with. To me that sounds like you are contemplating whether to reserve the right to try to screw over your colleagues at some later time. Unless that is your actual goal, I would steer well clear of this.
Yes, contracts matter, as they define the legal obligations between you and your employer. Note that it is not the contract only, but also the legal framework on which it is built (laws and decrees directly applicable to your situation). But, you have correctly identified the correct question to ask: even if the law is on your side, is it worth picking this fight? You can only answer by a careful analysis of risks and benefits. I list a few ideas below…
I have just been informed that my contract is research only.
Haven't you been handed your contract when you signed it? If not, that's a serious misconduct on their part and gives you leverage you can use.
I think technically this means I don't have to be teaching
This probably depends on the exact details of the contract, which we don't know. Read it and have it read by a lawyer, or at least a union representative or counselor.
Do I put up a fight about the change? To what extent do contracts matter?
What is missing most importantly here is your current relationship with your hierarchy (department chair, dean, whatever) and what is their position/wishes on this issue. Standing up to the bureaucrats is one thing, and while you may not get into their good graces, it can be a fight worth fighting. Standing up to your department chair can open a whole new can of worms.
This sounds like a potential big screw up. If your contract says one thing and you were promise another, you should put up a fight. Certainly, put up the fight before signing anything.
Contracts matter when someone tries to get you to do something you would rather not do. If you don't want a heavy teaching loads, then your research-only contract would help you avoid that. If you don't have such a contract, then you've lost your leverage.
Informal agreements do not stand the test of time, especially if there is a contradictory written agreement (signed by you).