Hi @Beckyxx and welcome to the forum. It is certainly possible that your remote is faulty but I suspect when you removed the paper tab some of it has broken off inside, which is quite common, and this will stop the batteries making proper contact.
Removing the back of the remote can be a problem the first couple of times as it it very stiff. I find holding the remote at both ends with the buttons down and my thumbs on top and then applying pressure and pushing with one thumb and pulling with the other in the direction of the arrow is sufficient to remove the cover. Note the back only moves a few millimetres and can then be lifted off.
A number of people have found that wearing rubber kitchen gloves helps to give more purchase on the plastic components.
The following is a troubleshooting guide for remote controls which might be of interest.
https://www.sky.com/help/articles/remote-control-help-sky-glass
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