MapThread a function with last result as input

Indeed, FoldList[] would be the appropriate operation here:

FoldList[{2, -1, 3, -2}.Append[#2, #1] &, 0, 
         Transpose[{{7, 1, 5, 10, 10, 2, 5, 10, 10, 6},
                    {4, 1, 4, 2, 5, 2, 2, 4, 8, 10},
                    {2, 6, 1, 4, 6, 7, 4, 1, 2, 8}}]] // Rest
   {16, -13, 35, -40, 113, -203, 426, -833, 1684, -3342}

---

Of course, modifying the snippet above to use fun[] in the OP is not too hard to do (spot the difference!):

FoldList[fun[Append[#2, #1]] &, 0, 
         Transpose[{{7, 1, 5, 10, 10, 2, 5, 10, 10, 6},
                    {4, 1, 4, 2, 5, 2, 2, 4, 8, 10},
                    {2, 6, 1, 4, 6, 7, 4, 1, 2, 8}}]] // Rest

An alternative to FoldList could be FoldPairList:

FoldPairList[
  Function[{prev, new}, {#, #}&@ ({2, -1, 3}.new - 2 prev) ],
  0,
  Transpose@sp
]

(* {16, -13, 35, -40, 113, -203, 426, -833, 1684, -3342} *)

---

In response to the request of keeping function fun definition, you can modify the above to obtain the same result as follows:

ClearAll[fun]
fun[{x_, y_, z_, a_}] := 2 x - y + 3 z - 2 a

FoldPairList[
  Function[{prev, new}, {#, #}&@ fun[{Sequence @@ new, prev}]],
  0,
  Transpose@sp
]

foldList = Rest @* FoldList[fun @* Flatten @* Reverse @* List] @* Prepend[0] @* Transpose;

Example:

sp = {{7, 1, 5, 10, 10, 2, 5, 10, 10, 6}, {4, 1, 4, 2, 5, 2, 2, 4, 8, 10}, 
  {2, 6, 1, 4, 6, 7, 4, 1, 2, 8}};

foldList @ sp

{16, -13, 35, -40, 113, -203, 426, -833, 1684, -3342}

Also

foldList2 = Transpose /* Prepend[0] /* FoldList[fun @* Flatten @* Reverse @* List] /* Rest

foldList2 @ sp

 {16, -13, 35, -40, 113, -203, 426, -833, 1684, -3342}