Dear friends,
I don't agree with you.
In main sentence there is "forget" not "remember" .
The main sentence is negative but you are offering a positive sentence.
As far as I know, 'don't forget' is used when you remind someone to do something while 'remember' is used when you say someone should keep something in their mind.
To paraphrase the sentence, keep in mind my home is always open to you. I don't know 'don't forget' can cover this meaning, but I'm certain about 'remember'.
And do not think of 'remember' as 'hatırlamak' here. It has a meaning like 'unutmamak'.