As far as the grammatical correctness goes for the two sentences that you've mentioned, they are both right. But if you mean to ask if they are both semantically equivalent and interchangeable, that's a no.
The dependent clause of the first sentence "even if sometimes it is raining" serves to emphasise on the first part of the sentence, "I take the dog for a walk every day". It reinforces the idea that whether it rains or not, the dog is taken out for a walk.
In the second sentence, however, "although" is helping point out an exception. That sentence may connote (aside from other possible meanings) that the dog is taken out for a walk on all days except on days when it rains. If you may be having difficulty in understanding how this may be so, try appending a little more context to that sentence. Consider this:
I take the dog for a walk every day although sometimes it is raining. On such days, we spend the day at home playing fetch.
However, there can be no way wherein the first sentence can be made to highlight an exception.
[edits: improvement in clarity, omission of 'but' in accordance with LPH's comment]