Hmm. Going after your intent, I'm going to lump all fossil fuel into this, and include coal and natural gas too.
I'm going to also assume that while the chemical engineers can turn almost any organic into any other organic for a price, we will consider that you really mean, "without breaking the bank"
Asphalt. Used in huge quantities, and the only real alternative to it is concrete, which is even more expensive. Asphalt, however is quite reusable, and increasingly you see them chip off 2 inches of road surface, then they reheat it mix in a small amount of rock and more asphalt, and re-lay it down.
Roofing tar. Not used in the same quanities, but not using a hot sealed roof would essentially require building a sloped roof on top of the roof.
Asphalt shingles. This does have an alternative -- fiberglass and various plastic shingles, and metal roofing. Metal roofing is a reasonable alternative. More expensive for the material, but it goes on fast. But building codes in many cities don't differentiate between the old galvanized 'rusty tin roof' and the powder coated easter egg colours of modern galvalum, or the textured metal shingles (not cheap) that are available now.
Tires
Plastics. There are degradable plastics coming out now (good thing) mostly made from corn starch, with all the sustainable issues of corn production. For one shot use this is a good idea. Lots of plastic however is longer term and you need materials that won't be eaten by your local neighbourhood bacteria or fungus.
Fertilizer. This one boils down to energy. Most nitrogen fertilizer starts as ammonia which in turn is made from natural gas. There are other processes for making ammonia, but they are more expensive.
Many of the things that use methane, (primary component of natural gas) use is as a source for hydrogen. All of these could use electrolytically split water for their hydrogen. Right now electricty is too expensive to use this way, but it may be a reasonable way to use surplus renewable electricity. One advantage of electrolysis is that it's easy to turn on and off, unlike many chemical processes.