Learn Danish: Hun beskylder mig for at lyve.

In Danish, verbs take specific prepositions that often don't match English. The verb at beskylde (to accuse) is always followed by for. So, while you say 'accuse of' in English, you must say beskylde for in Danish.

English uses the '-ing' form 'lying'. Why does Danish use the infinitive at lyve?

In English, prepositions are usually followed by the '-ing' form of a verb. Danish doesn't do this. When a preposition is followed by an action in Danish, you almost always use the infinitive marker at plus the basic infinitive form of the verb.

I learned that for at means 'in order to'. Is that what it means here?

No, this is just a structural coincidence! While the fixed phrase for at does mean 'in order to' in many sentences, here the two words just happen to be next to each other. The for belongs to the verb beskylder, and the at belongs to the verb lyve.

How do I remember not to confuse lyve with the verb for 'lying down'?

It is a classic trap because English uses 'lie' for both! In Danish, at lyve only means to tell a falsehood. If you want to say that someone is lying down or resting in bed, you must use the completely different verb at ligge.