E.g. How much longer are you staying in the country for your holiday?
残る sounds wrong. So is 住む。 Help appreciated.
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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？」.