How should I finish a wooden countertop that will have a stovetop and a sink?
Wood works great for countertops if you are willing to live with visible wear or even welcome the scars of daily use as part of the charme of a lived-in environment. 

You didn't mention which part of the world you're living in, so it's a bit difficult to recommend one wood over another - in general I'd go for something local that holds up well to rot and warp. In a rural environment, you might ask a farmer what they're using for (wooden) fenceposts. To name two less frequently mentioned choices that should be easily available in North America:
Black Locust (Robinia pseudoacacia) is very resistant to decay, and would hold up well in a wet environment. The same would be true for Osage Orange (Maclura pomifira), though the latter tends to come in smaller and gnarlier sizes. Depending on local availability, they might both be worth looking into.

About the woods mentioned by other replies: Maple in particular would be a wood I'd rather not use in that context, since it is not exactly known to hold up well to rot (and its bright colour and smooth grain would spell out any scruffs in all their glorious details). 

You have three sources of damage to your wooden countertops:


knives: Wood doesn't damage them, and sooner or later someone will forego whatever you're using as cutting surface and do the job directly on the countertop. With an oiled surface, these scars just add to the patina, and can be filled with wax or sanded off when the need arises. With a varnished surface you loose that advantage.
hot pans/pots leave visible scorch marks. Unlike fornica or sealed surfaces, these can be sanded off if so desired (provided the surface was oiled/waxed to begin with) and don't need to be considered damage.
water is your enemy #1. What may look like a neatly installed kitchen sink can grow ugly traces of rot around the edges if exposed to regular humidity. Keep in mind that you'll be splashing water liberally around a sink and faucet, and that stray jets of water will go splashing in any random direction once they hit the dishes piled up into that any-size-is-too-small sink. Add to that that the vicinity of the sink tends to clutter with tools, detergents, kitchen paraphernalia, dishes both clean(wet) and dirty, and forget all about "I'll just wipe it off when it happens".
For that reason, you will want your sink to be large enough to catch and contain most of the splashing.


From personal experience, you will want to keep both stovetop and sink installed in a way that keeps liquids from creeping into or under edges and heat sources at enough of a distance. I would aim for a flat, embedded installation, and make sure to use a flexible sealant for caulking. Depending on the wood used, the endgrain around the cutout could be primed with epoxy. This would have to be done prior to oiling the countertop.

Even more effective in the long run would be to reduce the amount of seams that need to be covered by choosing a modular sink that covers the belly-to-backsplash disaster zone with its own waterproof material without exposing wood nearby. 

There are many examples to be found across the web - just don't fall for the shiny interior-decorating-magazine pictures but imagine the setup after a decade of day-to-day use.

Distance of burner:
If you already settled on a burner - experiment in a safe setup which distance would be safe: Use a large pot to find out how far the flames will reach around, and make sure that radiated heat will remain low enough to avoid charring. If you have to install some kind of heat shield, keep in mind that the shield itself may get hot and cause the wood behind it to char unless properly insulated. 

Finish:


varnish: Not resistant to heat, will become damaged, exposing the wood. Looks great while new, and horrible after a few years of use. Not recommended.
PU coating: There are some products able to avoid damage. I don't know how well they hold up to heat. The result would basically be a PU-encased woodtop. I have not tried that.
Oil: Preferrably linseed oil (non toxic, food-safe): surfaces need to be re-oiled frequently (at least once a year, twice preferrable). This is less cumbersome than it might sound, though. Just rub/polish off any excess after oiling or it will take months to dry leaving you with a sticky half-dried surface meanwhile... Re-oiling also takes care of minor scruffs, larger ones can be sanded prior to oiling. Please bear in mind that rags used for oiling need to be closed off in an air-tight container to avoid self-incineration.