Selecting nonzero rules from a list

Your list is a list of rules. If you want to work with associations, do:

list = Association[a1->0,a2->3x,a4->0,a5->5x+3] 

<|a1 -> 0, a2 -> 3 x, a4 -> 0, a5 -> 3 + 5 x|>

Then, there are many ways to select the nonzero associations, here are a couple examples:

Select[# =!= 0&] @ list DeleteCases[list, 0] 

<|a2 -> 3 x, a5 -> 3 + 5 x|>

<|a2 -> 3 x, a5 -> 3 + 5 x|>

Use Join to append to another association:

Join[Association[b1->1, b2->3], Select[# =!= 0&] @ list] 

<|b1 -> 1, b2 -> 3, a2 -> 3 x, a5 -> 3 + 5 x|>

Addendum

On the other hand, if you do want to work with lists of rules (as mentioned in the comments), you could do:

list = {a1->0, a2->3x, a4->0, a5->5x+3}; Select[#[[2]] =!= 0&] @ list DeleteCases[list, _->0] 

{a2 -> 3 x, a5 -> 3 + 5 x}

{a2 -> 3 x, a5 -> 3 + 5 x}

Again, use Join to join two lists of rules:

Join[{b1->1, b2->3}, DeleteCases[list, _->0]] 

{b1 -> 1, b2 -> 3, a2 -> 3 x, a5 -> 3 + 5 x}