I think there are two relevant factors here.
Firstly, in an election where you are voting for a candidate to assume office, then a candidate who is found by the courts to have broken the rules in a substantive way will typically be disqualified from office. It's far less clear that that is a reasonable remedy when the person wasn't a candidate for office, but merely a prominent campaigner for one side or the other.
Secondly, invalidating an election because someone made an unrealistic promise (let's spend an extra £350m on the NHS, let's make Britain great again, let's abolish student loans) would surely invalidate every election there has ever been. Democracy can only work if we assume that the electorate treat such claims with a degree of scepticism. After all, it's open to the other side to challenge the claims.
However, I guess the question is about legal process: what could the courts decide to do? Well, it depends how creative they want to be. I've heard it argued that Article 50 requires due constitutional process to be followed before a state gives notice of withdrawal; a court could decide that due constitutional process was not followed and that the invocation of Article 50 was therefore null and void. That would put us into a major constitutional crisis. However, I find it hard to see a court coming to this conclusion, simply because the government did not constitutionally require a referendum in order to invoke Article 50; it was a decision it had the power to make anyway, and in that sense the referendum was legally irrelevant.