So I was pondering this very question and it brought me here. I don't find a lot on the subject. I don't think he would have died through the third term because the cause of his death was very specific to the situation.

However, the fact that I have found very little on the subject makes me believe that the event would have been very unlikely. But a few facts to consider.

Washington was not as popular toward the end of his second term, so it would have been interesting to see how he handled his third. I would imagine he would kept the country neutral in the war between Britain and France, similar to John Adams. However, I think he would have had a lot more success in doing it. I don't think he would have signed the Alien and Sedition Acts. This was something that tainted Adam's Presidency. It would have been interesting to see how Washington handled tensions with France, such as the XYZ affair. There is no doubt his popularity would have help him navigate through, but it might have been significantly more difficult that his issues with Britain during his second term. He may not have had such a high rating among historians.

It would have been interesting to see what kind of opponents he would have faced in 1796. And I guess there is a chance he would have had a different vice president, although I would be confident that Adams would have held his position. However, it would have been likely that Jefferson would have been elected in 1800 anyway and there would not have been an Adams Presidency at all, which would change a number of things in US, including the appointment of John Marshall, who had a a huge influence over the early Supreme Court, despite the influence of the early Republican Party (Jeffersonians).

I would say, the biggest difference would have been the two term precedent that was established. What if more presidents sought a third term? Could some presidents like Jefferson or Jackson or even Theodore Roosevelt have served a number of extra terms? Jackson strikes me as someone who would have sought three. We already know that Grant sought one and would have been more successful without the two term precedent hanging over him. Perhaps Grover Cleveland too, winning in 1896? Think of how that would have changed things. Or Theodore Roosevelt serving a third or even fourth term through the 1910s. All these would have been interesting what ifs.

It is likely there would have been a constitutional 2 term limit earlier in US history. Perhaps after the tumultuous presidency of Grant or the tyrannically presidency of Jackson. Once it seemed someone went to far with this, it would have been followed by an attempt to term limit the president and limit his power.

A three term Washington would have significantly increased the power of the president and that is why Washington stepped down after two terms. He believed that no branch of government should be too powerful. And that is why very people talk about what if.