Combustion, ventilation, and dilution air can be quite complicated. It is best to assume that the door ventilation is required and just replace it with a same louvered door.

The gas appliances require fresh air but most get it from a pipe from closet to attic. So no,the door doesn't have to be louvered if provisions are made to get air elsewhere.

" ...the general rule is that your furnace room cannot be smaller than 50 cubic feet per 1000 BTUs of heat the furnace produces. This almost certainly means that you will need to add a basic vent to your closet so that it can bring in a flow of new air for combustion."

furnace + water heater, and makeup air for a gas dryer

So, if I add a vent, I can replace the louvered door with a standard door?

Not necessarily. There are many factors, the size of vents you can get, where they go to, the size of your gas appliances. If it's worth removing the louvers, then you have to do the investigative work. Hiring out it's typically not something Joe HVAC technician knows or if does, wants to figure out just that one thing retroactively. Google IFGC 304 or IRC G2407 to get an overview.