Question: I realized I start sentences with *the*, *he*, *she*, *it*, *after* and *then* all the time. So I'm starting to use verbs instead. Here is an example of something I wrote:

> 
> Erin left out a sigh. "Why am I thinking so much about this?"
> 
> 
> She decided to forget about the thing, and go lie on bed instead. Talking to herself had really made her tired; a little bit more
> and she would go crazy. She reached to her desk and grabbed her phone.
> Hearing Benjamin's voice would probably snap her back to reality.
> 
> 
> 

I remember having read about this long ago (but I don't remember if the author considered it good or bad writing).

Answer: 
Begin a sentence with a verb sometimes. See, I just did. Twice.

As others have noted, you're apparently confusing gerunds with verbs in your example. But that said, gerunds are perfectly good words and can be used to vary your sentence structure. 

I don't know where you were told that a sentence can never begin with a verb. Imperative sentences (i.e. commands) routinely start with verbs: "Go to the store." "Bring me the book." Etc. It's relatively rare otherwise in English, as the conventional structure is subject-verb-object, but it's not unheard of. "Thinking quickly, Bob leaped for the door." "'Leaving so soon?' Sally asked." Sometimes it's effective to use a non-standard word order for emphasis or to save an unexpected word for last. "Left in the middle of the night, he did." "Kill me they might, but they will never defeat me."

If you find that you are in a rut and all your sentences follow the same pattern, then, as Tylerharms says, I would definitely make an effort to consciously break the pattern. (Now that you mention this, I have to look at my own writing and see if I've fallen into this!) Of course one should avoid the opposite extreme, of trying so hard to vary your sentence constructions that you over-use odd phrasings.
Every word, and how you use it are tools in your toolbox. You can follow proper grammar, or twist it to barely understandable, and these are still just tools.

What happens when you start your sentences with verbs? What is that tool used for?

The verb is the meat of the sentence. It's the most important word in the sentence. In fact if you don't have one, then you have a fragment. Other words in the sentence just describe the verb. Who is doing, the action, how is the action being done. Other even less important words describe the actors who are doing the action. The more flowery the writing, the more words you have besides the verb.

So now let's think about starting with verbs. We are jumping straight to the most important part. We don't have time to waste with other words. This makes the writing feel fast, direct, intense, personal. If those are the things you want, start with a verb. If they are not, then surround the verb with other words to give it the feeling that you need.

As always writing is best when it has ups and down. Sentences should be of varied lengths, and structure. You can't use any one tool for the whole piece of writing. Use many tools to create something interesting and changing, where the grammar itself is part of telling the story.