Question

In which sentence does the underlined infinitive phrase function as a subject? The goal of the new train station is to move more trains from the station to the city. The commuters were excited to ride on the new high-speed trains. To quickly move large groups of people is the purpose of public transportation.

Answer 1

The sentence in which the underlined infinitive phrase functions as a subject is the last one - to quickly move large groups of people is the purpose of public transportation. 
To quickly move is the infinitive phrase, and it is a subject.

Answer 2

The answer to your question would be that the sentence in which the underlined infinitive phrase functions as the subject is the following one: To quickly move large groups of people is the purpose of public transportation.

An infinitive clause is a subordinate clause whose verb is in this infinitive form.  Moreover, they are not introduced by a subordinating conjunction.