Suppose there is a planet fertilized with Earth-like life (characterized by proteins, nucleic acids, DNA, RNA) and a completely independent form of life, but also water and carbon based with different (or no) proteins and different mechanism of storing genetic information. So that the organisms of one form are not edible for the other form.
Will these two forms co-exist or one will eventually dominate or eradicate the other?
Yes.
Or no.
It depends.
Really you can plausibly set up whatever you like here.
If one form has a competitive advantage you can expect it to increase in numbers. But most likely some would find a niche where they have an advantage and occupy it.
So you could explain both or either happening quite easily but the most likely situation is that both would co-exist to some degree but with each dominating a different niche. It's interesting that A carnivores would not prey on B herbivores, and visa versa - so you may see a cyclical boom/bust situation where A is dominant for a while then their population crashes and B takes over until they also crash.
There would also be a strong evolutionary incentive to work out a way to consume each other.
Co-exist until they get to the Atomic Age
As the saying goes, no man is an island. Humans only survive because we're on top of the food chain and those in middle of the food chain only exist because of the bottom, all the way to zooplankton and the like. The entirety of Earth is custom built for our biochemistry.
That said, if a planet has two different types of biochemistry, then specialized areas will be developed for each one of them. And, due to the reactive nature of biological reactions, it's very likely that some of the normal compounds for Species A will be poisonous to Species B. Imagine trying to invade an enemy where the grass gives you horrible rashes, a grain of wheat eaten from their food stores will kill you, and their bacteria dies in your body but your body has no idea how to break it down. It's a logistical nightmare to the point that no Stone Age civilization would be able to hold an extended war against each other. So these live forms will live in a forced co-existence.
Until one of them gets technology, like heavier-than-air vehicles with sealed cockpits, heavy artillery with incredible range, rockets, and more. At that point, it's really hard to say. They could develop a peaceful co-existence, trading goods to each other that are universally valuable (like metals or information) and develop sterilized cities with which to meet each other. Or they wage war and nuke each other back to the Stone Age, which is more likely. Why? Well, because of the enforced isolation, ideas aren't easily traded, meaning that it'd be easy for a technology gap to be developed.
The two types of life will compete with each other and among themselves for resources.
Even though the two life forms have different biochemistries, they presumably need similar raw materials: reduced carbon for example, to build their bodies, water, reduced nitrogen, sunlight, etc. Just as on Earth creatures with similar biochemistries compete with each other, on your world creatures with different biochemistries will also compete with each other. Suppose I am an earth fungus in the soil, soaking up nutrients. Maybe some of the nonearth things are my neighbors in the soil, doing the same thing. Those jerks. If one of my kind evolves some toxin to poison them (as fungi use pencillin to poison their competitors the soil bacteria) there will be more of me and less of them.
Suppose I am a cow. My rumen turns out to be a nice place for nonearth microbes to live. Now if I eat some nonearth plants, my commensal nonearth microbes will break them down inside my body. Some of the resulting molecules are usable by my cow self, and my commensals get the rest. Or vice versa, with earthlife microbes acting as commensals for nonearth creatures.
The received knowledge for now is that archaebacteria are a different form of microbial life that was outcompeted by eubacteria around the time of the Great Oxygenation. There are still plenty of archaebacteria but they are now relegated places where oxygen is low or the environment is otherwise super harsh - salty, hot, etc. Maybe that will happen to one or the other type of life in your world.
Life will be life. Life forms will reproduce, evolve and compete.
