It is permissible to read athkar in bed. I usually finish fajr prayer, close the lights, go to bed, and read both the 'morning athkar' and 'sleep athkar' from my Athkar App on my iphone.
Is this ok to do? or are there rules when reading athkar?
There isn't any reason why it should not be permissible. One should do dhikr in all states.
الذين يذكرون الله قياما وقعودا وعلى جنوبهم
Who remember Allah while standing or sitting or [lying] on their sides ...
كان النبي صلى الله عليه وسلم يذكر الله على كل أحيانه
The Prophet ﷺ used to remember Allah in all situations.