Yes they will
Once the required link is in the cell, clicking on another cell makes the inserted link live - then clicking the link opens your browser. But the list of hyperlinks must be somewhere on the same sheet.
What I don't know is how to get the data validation part to link to the other sheet holding the hyperlink list.
Perhaps entering the data link manually would be possible, or use a VLOOKUP in the parent cell linking to the list sheet and cell refs or named range.