Infinitive or Gerund for celebration of an event?

Solution 1:

What you're confronting here is one of those fiendish composite verbal constructions English employs with relish to keep schoolchildren, their teachers, grammarians of every stripe, and speakers of non-English languages of the Wrong Sort^† in their respective places.

The construction here is [look forward to]. This must be carefully distinguished from the collocation [look (vb) + forward (adv.) + to (inf. marker)], which is how your first sentence must be construed:

My grandparents are looking forward to celebrate my arrival in February. = “My grandparents are directing their [figurative] gaze into the future, so that they may celebrate my arrival.”

Look forward to, employed as a fixed phrase, is a “phrasal verb”. It is transitive, taking a substantive — a noun, a pronoun, or a noun phrase — as its direct object. It may be glossed as “happily await” or “expect” or “anticipate”:

I look forward to Christmas. = “I await Christmas happily.”

There are two forms of a verb —celebrate, in your examples— which may act as a noun, and thus as the direct object of a verb such “to look forward to”:

• “to VERB”, which is called the *marked infinitive” because the bare infinitive of the form “VERB” is marked with the particle (not preposition) “to”, or
• “VERBing”, which when used this way is called a gerund because that’s what students of foreign languages of the Right Sort^† call it.

But even English has to draw the line somewhere — “look forward to to” is just too awful (in any sense) to contemplate. Consequently, look forward to takes the gerund, as in your second sentence:

My grandparents are looking forward to celebrating my arrival in February. = “My grandparents await, happily, celebration of my arrival in February.”

There's far more to parsing this insidious phrase; but you need not yet be troubled with the descriptive grammarians' cunning distinctions between prepositional and particular phrasal verbs. And you certainly don't want to be involved yet with the difficulties of transforming this sentence, into which you may be drawn if your editor is of an Enlightenment persuasion and deprecates ending clauses on a preposition because that's not done (nefas) in other languages of the Right Sort^†. It takes an iron determination (and a strong stomach) to come up with constructions like this:

We celebrate my arrival in February, forward to which my grandparents are looking.

In any case, the celebrate in your first sentence is not Future Simple, and the celebrating in your second sentence is not Future Continuous; they are, respectively, an infinitive and a gerund.

^† The Right Sort of non-English language embraces the Greek of Pericles, the Latin of Cicero, and possibly the Hebrew of Moses, who may be regarded as a British Israelite denied the opportunity to realize his Britannicism (just as was he prohibited from entering the Promised Land — in fact, the two may be the Same Thing). All other non-English languages are of the Wrong Sort.

Solution 2:

This sentence:

1. My grandparents are looking forward to celebrate my arrival in February.

That one has to do with forward-looking. It means that the grandparents are looking towards the future for the purpose of, or in order that, they might celebrate my arrival.

This sentence:

2. My grandparents are looking forward to celebrating my arrival in February.

This one means that they are anticipating the celebration of my arrival.

Only the second one really makes any sense here, because only here does the anticipatory sense of to look forward to something apply. The first is not a normal use, and risks being misunderstood.