The short answer is yes, you can use roofing felt under siding. It serves as a weatherproof barrier behind the vinyl siding and won't cause any issues if installed correctly. In some cases, you may need to use asphalt felt or two layers of grade D construction paper, which is commonly used on the West Coast. Polyethylene and polypropylene-based roof subfloors are also impervious to vapor and can be used as outdoor vapor retardants.
It's important to have a good air barrier and a vapor retardant inside, adequate ventilation and properly detailed gaskets on the ceiling, walls and windows. The building code (International Residential Code) requires that all new homes have a “water resistant barrier”, which in practical terms means an impermeable barrier behind the outer cladding. This creates an impermeable barrier similar to the scales of a fish; the overlay is large enough to direct water to the next layer and from the ceiling instead of upwards underneath an overlapping sheet. This is called “tile style” because it's the way roof and side wall shingles keep water out of the house.
Tar paper is distinguished from roofing felt, which is impregnated with asphalt instead of tar, but these two products are used the same way and their names are sometimes used informally as synonyms. You can use nails or washers from standard roofing covers (made of metal or plastic) to hold the felt paper in place until it is covered with the coating. Therefore, it is essential to cover the entire housing, from the roof to the foundation, including the ends of the gables and the girded joists, and always place the upper layers on top of the lower layers, in the form of shingles, to shed water. Felt, also known as underpayment, is required when asphalt shingles are installed as the first layer of roof or when applied to wooden shingles or a constructed roof.
Tyvek has existed, as a wrapper, since the 1970s (before it was mainly used for break-proof envelopes). In fact, I just installed a new roof and asked for ASTM type 15 felt instead of the synthetic one they now prefer. Building codes vary from region to region, and most building codes require a felt or synthetic subfloor for roofing. When installing felt paper under siding, it's important to make sure that it's done correctly. You'll need to ensure that you have an impermeable barrier behind your outer cladding that will direct water away from your home.
You'll also need to make sure that you have a good air barrier and vapor retardant inside your home as well as adequate ventilation and properly detailed gaskets on your ceiling, walls and windows. You'll also need to use nails or washers from standard roofing covers (made of metal or plastic) to hold your felt paper in place until it's covered with your coating. Finally, make sure that you cover your entire home from roof to foundation including all gables and girded joists.