My question is that if such a spring has mass, then to produce tension it, will the forces that are to be applied on the opposite sides of it be equal , without causing any acceleration in the spring as a whole.
I think that the answer should be that such forces would not be equal, because if such a string is tied to a wall on one end (A) and is pulled from the other (B), then it will have different values of tension at different points, the highest being at A and lowest being at B. Using that analogy we may be requiring different forces at different endpoints of the string.
The above reasoning also implies that you could apply a greater force from either end and the lower force will simply apply at the other end.
In opinion of friends at my locality, the forces should be equal just like in the case of massless string.
Many high school kids relate to this , so this is a legit question.