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I mean I don't know what country you are asking about, but it doesn't really matter where you live, the Rules of the House are available to anybody, right there on the internet, so anybody who's interested can see for themselves how this process works.
Democrats don't need to vote for the Republican choice, but so long as they vote for their own choice instead of answering "present" they are adding to the number of votes that need to be overcome to move forward.
Each representative has to decide for himself whether keeping the House shut down is the right way to go, and we see how they are voting unanimously, helping the fringe Republicans in their bargaining position.
