What is the difference between I don`t forget walking along the river and I remember walking along the river ?Feel free to just provide example sentences.

Both mean the same thing, except "I don't forget walking along the river" makes no sense, it's not grammatically correct, the tenses are mixed up. "Don't" implies a more present/future tense, for example "Don't turn on the television" or "Don't eat the cake".
So if you want to say it in that sense, you wouldn't use "don't", you could use "didn't" but that would change the whole point of what you're trying to say because I'm guessing you're trying to say that you recall walking along the river. If you used "didn't", it would imply that you had to do something (walk along the river) and you remembered to do it, and in order to make that grammatically correct, you would also change "walking" to "walk"
— "I didn't forget to walk along the river".

Long story short, as you're saying that you've walked along the river before, and you remember it, you would say "I remember walking along the river".