When do I place an article before a verb?

The short answer to your question is never.

Articles are elements of noun phrases, not of verb phrases. Therefore they cannot be used with verbs, only with nouns.

1. This gizmo is responsible for opening the door.

2. This gizmo is responsible for the opening of the door.

In sentence (1), you have a verb phrase so you do not use an article. You know it is a verb phrase because it has a direct object: the door. We do not use prepositions to connect a verb’s object arguments to that verb. However, because this particular verb phrase is acting as the prepositional complement for the preposition for, some folks also refer to it as a gerund clause. In any event, articles are forbidden at the start of verb phrases like this.

In sentence (2), the all-important addition of the preposition of to connect the -ing word to the door following it means that that door is no longer the direct object of any verb, just the second of two nouns connected by a normal preposition. Because now it is a noun phrase, the article is allowed.