是have fun doing sth.不是have fun to do sth.
have fun doing sth.实际上是have fun in doing sth.在这个短语中，介词in可以保留，也可以省略。
类似的结构还有：
1.have trouble (in) doing sth.
We had no trouble in finding his house.
2.have difficulty (in) doing sth.
I have no difficulty in working out this problem.
3.have problem (in )doing sth.
Mary had problem in finishing much work by herself.
4.have a good time (in) doing sth.
The children are having a good time in playing basketball.
……
首先先回答楼主的问题，have fun doing something和have fun to do something都是有的。
下面我来解释一下两个的区别have fun doing和have fun to do的区别为：
一、指代不同
1、have fun doing：玩得开心。
2、have fun to do：做某事会开心。
二、侧重点不同
1、have fun doing：doing表示的就是开心的这件事。
2、have fun to do：动词后跟一个不定式表示一种目的。
三、引证用法不同
1、have fun doing：fun是不可数名词，没有复数形式，也不可加不定冠词，其后可接(in) v -ing。fun可用于另一名词前作定语。
2、have fun to do：fun普通用词，指任何能给予喜悦的娱乐，或指娱乐本身。
