How to programmatically modify a function definition

This is my first time wading into metaprogramming in Mathematica, so take this with a pinch of salt. I can get the DownValues of myfunction and strip out the cases of Print commands, then build a new function newfunction by setting DownValues like so:

(* any old function will do *)
myfunction[x_, y_] := Module[{p = 0},
  p = x^2 + y^2;
  Print["test1: " <> ToString@p];
  If[p < 1, p = p^2 + 1, p = y - x];
  Print["test2: " <> ToString@p];
  Do[
   Print["blah" <> ToString@i];
   , {i, 3}];
  Return[p]
]

DownValues[newfunction] = ReleaseHold[
   DeleteCases[
     DownValues[myfunction][[1]],
     fn_[___] /; fn === Print, Infinity,
     Heads -> True
     ] /. myfunction -> newfunction
   ];

myfunction[6, 3]
(*
> test1: 45
> test2: -3
> blah1
> blah2
> blah3

returns -3
*)

newfunction[6,3]
(* returns -3 *)

You might want to look into ways to suppress Print though, because the above technique looks pretty dangerous and will probably go wrong in unexpected ways. Suppress Print[ ]s?

To operate on notebooks

Let nb be the notebook you want to alter obtained with NotebookGet[]. For instance, it could be nb = NotebookGet[EvaluationNotebook[]]. Instead of EvaluationNotebook[], you could have something like First@Select[Notebooks[], Information[#, "FileName"] === "MyProg" &].

nb /. HoldPattern@RowBox[{
      x___, Optional[";", ";"],
      RowBox[{"Print", "[", ___, "]"}],
      Optional[";", ";"], y___}] :>
   RowBox[{x, y}] // NotebookPut

Note: This will not extensively tested. Like the method below, it may result in errors in the code. It should work well if each Print[] statement occurs in a CompoundExpression.

To operate on definitions in the kernel

The function cleanup[sym, pat] will delete all expressions matching pat from the definitions of a symbol sym. Use _Print to delete Print statements.

cleanup[sym_Symbol, pat_] := 
 Language`ExtendedFullDefinition[sym] = 
  DeleteCases[Language`ExtendedFullDefinition[sym], pat, Infinity]

Deleting Print[..] as an argument to something other than CompoundExpression may result in errors on execution. For example:

DeleteCases[Hold[Module[{}, Print["Hi there!"]]], _Print, 
  Infinity] // ReleaseHold

Module::argmu: Module called with 1 argument; 2 or more arguments are expected.

(* Module[{}]  *)

Adding a semicolon, Module[{}, Print["Hi there!"];], prevents the error.

Example

Example function to clean up, showing a variety of values (DownValues, SubValues and UpValues):

ClearAll[addto];
call : addto[x_, y_] := (Print["main routine called: ", 
    HoldForm[call]]; x + y);
call : addto[x_][y_] := (Print["operator form called: ", 
    HoldForm[call]]; addto[x, y]);
addto /: call : 
   addto[x_] + y_ := (Print["upvalue form called: ", HoldForm[call]]; 
   addto[x, y]);

Test:

addto[3][4]

operator form called: addto[3][4]

main routine called: addto[3,4]

(*  7  *)

addto[3] + 5

upvalue form called: 5+addto[3]

main routine called: addto[3,5]

(*  8  *)

cleanup[addto, _Print]

(*
Language`DefinitionList[HoldForm[addto] -> {OwnValues -> {}, 
   SubValues -> {HoldPattern[call : addto[x_][y_]] :> 
      CompoundExpression[addto[x, y]]}, 
   UpValues -> {HoldPattern[call : addto[x_] + y_] :> 
      CompoundExpression[addto[x, y]]}, 
   DownValues -> {HoldPattern[call : addto[x_, y_]] :> 
      CompoundExpression[x + y]}, NValues -> {}, FormatValues -> {}, 
   DefaultValues -> {}, Messages -> {}, Attributes -> {}}]
*)

Test again:

addto[3][4]

(*  7  *)

addto[3] + 5

(*  8  *)

That's not a real answer... But I failed to format it as a comment. Again.

You're opening a deep can of worms now, called "metaprogramming".

Please search for "metaprogramming" and discover excellent posts by Leonid Shifrin and others.

In your particular case:

increment = Function[{x}, Print[x]; x + 1]

You can try something like this:

increment //. { 
  HoldPattern @ CompoundExpression[a___, _Print, b___] :> 
   CompoundExpression[a, b]} 

The original increment has a tree representation:

The result will look like:

Generally speaking you can use rewrite rules to manipulate the structure of the expression (most of the times you will have to inactivate the expression with 'Hold' and friends)