The two sentences you presented at the outset are both grammatically and semantically correct, so you can write them any which way you prefer.
The first sentence, “I’m committed to providing whatever it takes to meet a students’ need,” uses the gerund phrase “providing whatever it takes to meet a students’ need”—which, of course, is a noun form—as the object of the preposition “to.” Through the agency of this preposition, that gerund phrase “receives” the action of the verb “committed.”
On the other hand, the second sentence, “I’m committed to provide whatever it takes to meet a students’ need,” uses the infinitive phrase “to provide whatever it takes to meet a students’ need”—also a noun form—as a verb complement of “committed.” This verb complement functions both as a modifier of the verb “committed” as well as its direct object, meaning that it directly receives the action of that verb. It’s clear that this alternative sentence construction is, like the first, grammatically beyond reproach.
However, by some quirk of the English language (an aspect so abstruse that I will not attempt to explain it here), not all verbs will accept an infinitive phrase as its verb complement. This is the case with the verb “see” in this awkward form of the second sentence you provided: “I look forward to see you for public school next year.” The construction may appear to be somewhat grammatically correct but it just doesn’t sound right. But it would definitely sound grammatically correct and idiomatic if we use the gerund phrase “seeing you for public school next year” as the object of the preposition “to”: “I look forward to seeing you for public school next year.”
For the same reason, this construction of the third sentence you presented, “‘The CHED appears to have conducted the GTS last year, and I look forward to examining the results,’ he said,” is grammar-perfect and sounds right as well. Here, the gerund phrase “examining the results” is the object of the preposition “to.” In contrast, the construction “‘The CHED appears to have conducted the GTS last year, and I look forward to examine the results,’ he said” looks grammatically flawed and sounds awkward as well. Here, the infinitive phrase “to examine the results” functions as a verb complement of the verb “look”—a construction that on the face of it just doesn’t work right.
In all the constructions above, the words “providing,” “seeing,” and “examining” function not as verbs but as gerunds—verbals ending in “-ing” that, of course, function as a noun form; the “to” that precedes them doesn’t make them infinitives but is simply a preposition that links them to the object of the preposition. In the alternative sentence constructions, on the other hand, the words “to provide,” “to see,” and “to examine” function as infinitives—verbals consisting of the base verb preceded by “to” that likewise function as a noun form. In the sentences concerned, the gerund phrase that follows each of these infinitives functions as a verb complement (a modifying phrase), not as object of the verb.
