Is there a Mitzvah of Tzitzis at night? If not, why do some people wear them at night?
This is the shittah of the Rosh, who holds that a beged yom is chayav even at night. I believe the Mishnah Berurah also brings down from the Arizal to wear tzitzis at night al pi kabbalah.
The Shulchan Aruch in O"H 18:1 quotes a machloket rishonim on how to interpret this gemara:
Rambam: At night, you are patur from wearing tzitzit on your garment, regardless of what kind of garment it is.
Rosh: If the garment is specialized for daytime wear, you always need to have tzitzit on it. If the garment is specialized for nighttime wear, you never need to have tzitzit on it, regardless of whether it is currently day or night.
I recall a Tosafos in Yevamos in the sugya of esay doheh lo taaseh that proves that there is one shittah in the mishnah that holds tzitzis to be obligatory at night. Perhaps it is a stringency to fulfill that opinion.
No (Rabenu Bachya and other Mefarshim to Debarim 15:39).
there is no mitzvah to wear tzitzis at night time. tzitzis is a mitzvah which has a specific time to be fulfilled during. This is part of the reason women don't wear them b/c women are not obligated in those mitzvas which are time dependent.
Even though it is not a mitzvah at night we don't require someone remove their tzitzis. Additionally like the kitzur quotes from tehillim I have set Hashem before me always and since the mitzvah of tzistzis is so fundamental to reminding of us of G-d one should be careful to even wear at night when the biblical mitzvah no longer applies.
Lastly there is a custom although not universally accepted b/c of disagreements concerning blessings to also wear a tallis katan to sleep. Many are careful to have a set of tzitzis to wear to sleep as well so that way if when gets up from sleeping to use the bathroom or something else then one is still wearing tzitzis when walking around.