Poly time superset of NP complete language with infinitely many strings excluded from it
Every $\mathsf{coNP}$-complete set contains an infinite subset in $\mathsf{P}$ assuming that 


pseudorandom generators exist, and 
secure one-way permutations exist.


In other words, assuming that these two conjectures are true, 
no $\mathsf{coNP}$-complete set is P-immune.
As pointed out in the comments by Lance, 
this is implied by Theorem 4.4 of 


Glasser, Pavan, Selman, and Sengupta, "Properties of NP-complete sets",
SIAM J. Comput. 36(2), 516–542.


(Kaveh has already shown that your question is equivalent to 
whether every $\mathsf{coNP}$-complete sets 
contains an infinite $\mathsf{P}$ subset. 
In other language, 
this is saying that no $\mathsf{coNP}$-complete set is "$\mathsf{P}$-immune." This is the language used in the above-referenced theorem.)