That could potentially cause them problems, yeah, if you were able to do that endlessly. In practice most companies will have some kind of rate limiting in place around features like that (by IP, cookie, captcha, etc.)
Together with cookies, you can show the captcha only to visitors that are not already recognized in some way, giving them a limited number of actions before showing the captcha. And regardless of whether you want one on your password reset page, you almost certainly want one on your login page anyway.