E.g. How much longer are you staying in the country for your holiday?
残る sounds wrong. So is 住む。 Help appreciated.
In general, when asking how long someone will stay in a certain place, one would use 居る{いる}.
住む implies long-term (permanent) residence. 残る would imply remaining beyond what was expected or allowed as a length of stay. 居残{いのこ}る = To stay behind, to be left behind. Even if you are asking someone how much longer they will remain, you would still normally use:
「あとどれぐらい日本にいますか?」(How much longer will you be in Japan?)
For stay at accommodations specifically, 滞在{たいざい} would be appropriate.
As was mentioned in one of the comments, people will often just ask when you are returning to 'your' country, à la 「国にはいつ帰{かえ}るの?」.
いつ帰るの?
)