I want to make a sentence like
I feel like I am on pins and needles.
Then, am I right to write like
I feel on pins and needles.
Can I write the sentence without “like” part?
Your example sentence: 'I feel on pins and needles' is not correct English. You could instead use the sentences "I am all pins and needles" or "I am on pins and needles".
The sentence 'I feel like I am on pins and needles' is a sentence that is using an English language 'idiom'. If you used the sentence "I feel pins and needles", this would break the idiom and indicate that you are actually touching pins and needles.
Idioms are sentences or phrases that have a specific meaning that often doesn't match what the words in them would imply. This specific idiom ("on pins and needles") describes a condition where someone is very nervous or excited because of some anticipated event (and not literally being on top of pins and needles). Normally you can't change any words in an idiom without destroying the specific meaning of that idiom.