There are mixed answers online. Some say 't' is silent. Others say 'e' is silent.
Dictionaries don't seem to give a consistent pronunciation for this word, so it's hard know what is correct.
The < t > is always silent. Whether the < e > is silent or not depends on whether the speaker uses a syllabic /n/ at the end of the word or, alternatively, uses the schwa vowel plus an /n/. The two choices are : /fɑ:sn̩/ and /fɑ:sən/.
You may see either pronunciation in a given dictionary. For example, if you stick fasten definition into a Google search, you will get a definition from Oxford Dictionaries alongside the transcription "/fɑ:sn/" with no mention of a possible schwa before the /n/.
However, a good dictionary will show that both pronunciations are possible. This is usually shown by putting a small, superscript schwa symbol < ə > before the /n/:
Note that this does not mean that there is a small schwa sound before the /n/! It means that the schwa is optional. Sometimes it will be there and other times it won't.
It's probably worth mentioning that the version with the schwa may be thought of as a slightly child-like pronunciation.