diff --git a/lib/matplotlib/axes/_base.py b/lib/matplotlib/axes/_base.py
@@ -3261,7 +3261,12 @@ def _update_title_position(self, renderer):
             if title.get_text():
                 for ax in axs:
                     ax.yaxis.get_tightbbox(renderer)  # update offsetText
-                    if ax.yaxis.offsetText.get_text():
+                    # A hidden offset text (e.g. on the shared y axis of an
+                    # inner subplot) is not drawn, so it must not move the
+                    # title: its tight bbox is non-finite and would otherwise
+                    # push the title to infinity.
+                    if (ax.yaxis.offsetText.get_visible()
+                            and ax.yaxis.offsetText.get_text()):
                         bb = ax.yaxis.offsetText.get_tightbbox(renderer)
                         if bb.intersection(title.get_tightbbox(renderer), bb):
                             top = bb.ymax

diff --git a/lib/matplotlib/tests/test_axes.py b/lib/matplotlib/tests/test_axes.py
@@ -8009,6 +8009,22 @@ def test_title_above_offset(left, center):
         assert ycenter == yleft
 
 
+def test_title_above_hidden_offset():
+    # On an inner subplot with a shared y axis the offset text is hidden, but
+    # it still carries text. It must not be considered when positioning the
+    # title: its tight bbox is non-finite and used to push the title (and the
+    # subplot position computed by tight_layout) to NaN/inf. See #31881.
+    mpl.rcParams['axes.titley'] = None
+    fig, axs = plt.subplots(1, 2, sharey=True, tight_layout=True)
+    for i, ax in enumerate(axs):
+        ax.set_title(f'Subplot {i}')
+        ax.plot(range(10), [1e53] * 10)
+    fig.draw_without_rendering()  # used to raise ValueError (NaN -> int)
+    for ax in axs:
+        assert np.isfinite(ax.title.get_window_extent().ymin)
+        assert np.all(np.isfinite(ax.get_position().bounds))
+
+
 def test_title_no_move_off_page():
     # If an Axes is off the figure (ie. if it is cropped during a save)
     # make sure that the automatic title repositioning does not get done.