stock will go down as he needs to sell a portion to pay taxes.
As usual, there are competing forces at play here.
First, forget that it is Elon, or any other huge shareholder, and just look at the effect on the company of Joe Random exercising options. A small amount of cash goes to the company to pay for the shares at the strike price. At the same time an obligation by the company to issue the shares gets retired. So the company balance sheet looks a little bit healthier. This is offset by the dilution of the stock by a few new shares. I have heard from others much more knowledgeable than me that the net effect is slightly positive, the company looks more stable.
When one exercises the options one presumably owes income tax (since you wouldn't exercise unless they were in the money). The difference between the strike price and the market price is treated as income,
but you have to pay the tax immediately or the transaction can't proceed. For most people this is transparent; the broker actually lends you the money to exercise and pay the tax, they do the exercise, then they sell some of the stock to pay the themselves back for the loan. Unless the person has a LOT of options, this is unlikely to move the market either way.
If you are Elon, you could pledge the soon-to-be-owned, or already owned, stock toward a loan, and use that money to pay for the exercise and the tax, and just keep all the shares. This would be dilutive, and maybe enough to depress the stock all by itself. But historically Elon doesn't seem to have done that, instead dumping stock to fund the exercise, and that definitely tends to tank the stock.