You're supposed to use a gerund after a preposition. e.g., Carry on running!

Yes, as long as you are adding a verb after "carry on" as a direct object, you need to put it in the gerund "-ing" form. This is because "carry on" requires a nominal (noun) object. E.g.: He carried on *the conversation* ("the conversation" is a noun.) She carried on *the family business* Now, when you have a verb that follows "carry on" you nominalize it (make it into a noun) by using the gerund form of the verb, by adding -ing. E.g., He carried on his singing. The doctor carried on caring for his patients despite being sick himself. Of course, if the verb is no longer a direct object of "carry on," then you do not necessarily have to use the gerund "-ing" form. E.g., He carried on as if nothing happened.