diff --git a/CHANGES.rst b/CHANGES.rst
index 686ff02f80..fc3ef437db 100644
--- a/CHANGES.rst
+++ b/CHANGES.rst
@@ -14,6 +14,8 @@ Unreleased
     If subclasses were overriding these methods, the old signature is detected,
     shows a deprecation warning, and will continue to work during the
     deprecation period. :issue:`5815`
+-   The ``should_ignore_error`` is deprecated. Handle errors as needed in
+    teardown handlers instead. :issue:`5816`
 -   ``template_filter``, ``template_test``, and ``template_global`` decorators
     can be used without parentheses. :issue:`5729`
 -   ``redirect`` returns a ``303`` status code by default instead of ``302``.
diff --git a/src/flask/app.py b/src/flask/app.py
index 0fc927289b..c17da4b49d 100644
--- a/src/flask/app.py
+++ b/src/flask/app.py
@@ -995,6 +995,17 @@ def full_dispatch_request(self, ctx: AppContext) -> Response:
 
         .. versionadded:: 0.7
         """
+        if not self._got_first_request and self.should_ignore_error is not None:
+            import warnings
+
+            warnings.warn(
+                "The 'should_ignore_error' method is deprecated and will"
+                " be removed in Flask 3.3. Handle errors as needed in"
+                " teardown handlers instead.",
+                DeprecationWarning,
+                stacklevel=1,
+            )
+
         self._got_first_request = True
 
         try:
@@ -1576,7 +1587,11 @@ def wsgi_app(
             if "werkzeug.debug.preserve_context" in environ:
                 environ["werkzeug.debug.preserve_context"](ctx)
 
-            if error is not None and self.should_ignore_error(error):
+            if (
+                error is not None
+                and self.should_ignore_error is not None
+                and self.should_ignore_error(error)
+            ):
                 error = None
 
             ctx.pop(error)
diff --git a/src/flask/sansio/app.py b/src/flask/sansio/app.py
index 0195b16cd7..e069b9087a 100644
--- a/src/flask/sansio/app.py
+++ b/src/flask/sansio/app.py
@@ -922,15 +922,16 @@ def trap_http_exception(self, e: Exception) -> bool:
 
         return False
 
-    def should_ignore_error(self, error: BaseException | None) -> bool:
-        """This is called to figure out if an error should be ignored
-        or not as far as the teardown system is concerned.  If this
-        function returns ``True`` then the teardown handlers will not be
-        passed the error.
+    should_ignore_error: None = None
+    """If this method returns ``True``, the error will not be passed to
+    teardown handlers, and the context will not be preserved for
+    debugging.
 
-        .. versionadded:: 0.10
-        """
-        return False
+    .. deprecated:: 3.2
+        Handle errors as needed in teardown handlers instead.
+
+    .. versionadded:: 0.10
+    """
 
     def redirect(self, location: str, code: int = 303) -> BaseResponse:
         """Create a redirect response object.