As Tyler James Young has suggested, using a double negative in order to say something affirmative, goes as follows:
"There aren't any reasons not to do it."
Moreover, as Barrie England has suggested, one mustn't change the infinitive when double-negating. Think of your sentence in the following way:
"To do it, there are no reasons not."
With completely different sentences, the same rule applies. For example, with to apply,
"It's not that he didn't refuse to apply himself, it's just that his heart wasn't in it."
(i.e., "Not to apply himself, he did not refuse, it's just that his heart wasn't in it")
Or with the infinitive to boast,
"May it be not far me to boast not about him when he is in my presence." (i.e., "To boast not about him, may it not be far from me, when he is in my presence.")
That sort of thing. In some ways, double negation is a kissing cousin to litotes, the figure of speech which uses the negation of the positive to indicate something positive (There was no small stir when the unpopular decision was made. In other words, there was a big ruckus when the unpopular decision was made.)
Both litotes and double negation can be used artfully, but in using them you run the risk of being misunderstood!