Grammar 'more' and 'faster'

No, it is grammatically wrong if you put more before faster in this sentence.

This is a common mistake for comparative and superlative adjectives.
We add "er" to compare 2 items and "est" to compare 3 or more items
eg.
Small
1. An apple is smaller than a watermelon. (comparing the size of apple and watermelon, only 2 items)
2. Among lemon, apple and watermelon, lemon is the smallest. (comparing the size of lemon, apple and watermelon, 3 items)

However, the rule changes when the adjective has 3 or more syllables (sound of the word)
eg. Expensive (the word read as ex-pen-sive, 3 sound) or Convenient (read: con-vin-nei-ant, 4 sound)
So in this case, we will not add "er" or "est" to them, instead, we add "more" or "most"
eg.
1. My shoes are more expensive than Tim's. (comparing between you and someone)
2. I have the most expensive shoes in my class. (comparing among your and the rest of your class)

Let's get back to your sentence:
Fast, only 1 syllable, so we will simply add "er" to it.
Peter can run faster than John.

No, faster is a comparative form of the adjective fast. More is redundant in your sentence. However, you can say "much faster", "a lot faster", or in N. America, "way faster".

Peter can run more faster than John.
In comparative and superlative forms of some of these words,eg:-faster, are used because it is considered in its literal sense:-"can run more faster" really means:-"can run faster..."
If the superlative without most is used,and the comparative more should not precedes another comparative"faster".eg:-
can run faster than.......would be enough.