Question

In which sentence is the underlined infinitive or infinitive phrase functioning as a direct object?

Answer 1

It is B, you are finding the infinitive as a direct object, in the answer B "to drink water" is the infinitive.

Answer 2

The infinitive phrase is functioning as a direct object in B.

(b) After all that exercise, the runners needed to drink water.

Take the verb "to need". This transitive verb takes two arguments, one that functions as the NP subject of the sentence and denotes the one undergoing the need ("the runners")  and an NP direct object depicting what the subject needs ("to drink water").

In A the infinitive phrase functions as the subject of  the sentence ("to run in a marathon") and the same applies in D ("to cheer on the runners"). In C, the infinitive phrase ("to give an award") functions as the complement of the preposition about.