Could someone running to replace Cameron as PM just pledge not implement Brexit vote result as part of their platform? The Brexit vote is non-binding, so wouldn't a PM candidate campaigning with the promise of setting aside the outcome of the Brexit somewhat legitimize not implementing the referendum's result?
The simple answer is yes.
However, there are a couple of important aspects that need to be considered.
The next Prime Minister will come from within the Conservative MP block and they are nominated by MPs from within the Tory Party, ratified by the Tory executive, and in the case of more than one nominated candidate, elected by the Tory Membership. The accepted answer to this question details the process in more detail.
If there is more than one candidate, and it goes to a vote through the membership, then the 'platform' on which a candidate stands would geared towards the Tory Membership. Now a breakdown of how Tory voters voted in the referendum shows that they voted to Leave by 58% to 42%, assuming the membership followed a similar pattern. Therefore, it would be unlikely that a candidate would openly campaign to obtain a mandate to not enact Article 50.
The other aspect to consider is that the UK's membership of the EU is enshrined in law. To enact Article 50 would require the repealing of that law and this requires an Act of Parliament. Therefore a Prime Minister cannot enact Article 50 without the MPs ratifying that decision. And this where it can get interesting as the majority of the Members of Parliament were aligned with the Remain campaign. If enough of them (majority on the motion) provide a vote of 'no confidence' in the current Government then a General Election would be called. There are 23 examples of a successful motions of no confidence in a UK Government; the last being in 1979 when the Labour Prime Minister, James Callaghan, dissolved the Government after the 'Winter of Discontent'.
With the Government dissolved Article 50 would remained un-enacted until a new Government is formed.
This would mean a General Election and campaigning on a national scale. If the next Prime Minister, and their Government, was voted in on a 'Remain' campaign pledge they would have a legitimate mandate to ignore the referendum and not enact Article 50.
Yes, de iure, it is possible for the future prime minister to be someone who is determined not to pay attention to the referendum. Because the U.K. Parliament picks the prime minister, it would almost certainly have to mean that a majority of the lawmakers finds this attitude acceptable, too. The only problem is that this politician and those who adopt his or her attitude are flagrantly dismissing the will of the people. It's something not expected in the U.K. which is one of the cradles of democracy, freedom, and capitalism.
On the other hand, David Cameron is already helping this scenario. The very fact that he wants to avoid the "activation of Article 50" – he doesn't want to send the letter to the European Commission saying that the departure process is being officially started and he wants his successor to do it – means nothing else than the fact that Cameron wants his successor to have the freedom to decide whether the letter is sent at all.
If Cameron thought that the successor shouldn't have this freedom to acknowledge or dismiss the referendum, he would begin the departure process himself. So it's possible that after some time, it becomes "politically acceptable" to ignore the referendum and Cameron's successor will actually do so. It would clearly mean that Cameron's pledge of the referendum and the referendum itself will have been a fraudulent game. To some extent, it's already now. If Cameron were decent and respectful towards the referendum, he would send the letter himself and either start the process of the departure, or immediately resign to allow someone else to do the work.
One must realize that the departure of the U.K. has the other side as well. The EU will try to influence things in various ways. I think that the most important EU officials have understood that a majority of the Britons isn't really compatible with the kind of the EU they want to build. So politicians like Juncker probably actively want the U.K. to leave. But as discussed in a previous question, they have no legally prescribed procedures to expel a member state. The referendum result means nothing from the perspective of the EU law – only the letter activating Article 50 would.
It's totally plausible that the U.K. will continue to be a full member for a long time with the extra advantage that it may send the "Article 50" letter at any moment and it may use this threat as a powerful weapon. This situation is clearly a defect in the EU law that no one has been prepared for. But there are many other, more serious defects in the EU law so one shouldn't overstate the importance of this particular glitch.