If you've seen "I don't mind to get up early" (yes, I've found a couple of examples), then the writer has made a mistake. That's all.

You might have seen a "from / to" phrase with a gerund where "to" is a preposition and V+ING is more like a noun. In London you can do anything from eating breakfast in bed to drinking at night. With verbs "to" is normally part of the infinitive form "to run, to walk etc" Prepositions are a nightmare, hopefully someone on here will explain this better than me.

Do you like getting up early / like to get up early? to enjoying - nope to stopping - nope to minding - nope to finishing - you getthe idea. to suggesting - see above. Do you like to get up early? Do you like getting up early? Do you like getting up early and eating breakfast? Do you like getting up early to go for a run? [to go for a run] is a separate clause, not part of "like getting"

Example? Sorry, not sure what you are asking.