diff --git a/pandas/tests/indexes/common.py b/pandas/tests/indexes/common.py
index f3ebe8313d0c6..dbfa39ef690f5 100644
--- a/pandas/tests/indexes/common.py
+++ b/pandas/tests/indexes/common.py
@@ -6,6 +6,7 @@
 
 from pandas._libs.tslib import iNaT
 
+from pandas.core.dtypes.common import is_datetime64tz_dtype
 from pandas.core.dtypes.dtypes import CategoricalDtype
 
 import pandas as pd
@@ -301,6 +302,9 @@ def test_ensure_copied_data(self, indices):
 
         index_type = type(indices)
         result = index_type(indices.values, copy=True, **init_kwargs)
+        if is_datetime64tz_dtype(indices.dtype):
+            result = result.tz_localize("UTC").tz_convert(indices.tz)
+
         tm.assert_index_equal(indices, result)
         tm.assert_numpy_array_equal(
             indices._ndarray_values, result._ndarray_values, check_same="copy"
@@ -464,6 +468,11 @@ def test_intersection_base(self, indices):
         intersect = first.intersection(second)
         assert tm.equalContents(intersect, second)
 
+        if is_datetime64tz_dtype(indices.dtype):
+            # The second.values below will drop tz, so the rest of this test
+            #  is not applicable.
+            return
+
         # GH 10149
         cases = [klass(second.values) for klass in [np.array, Series, list]]
         for case in cases:
@@ -482,6 +491,11 @@ def test_union_base(self, indices):
         union = first.union(second)
         assert tm.equalContents(union, everything)
 
+        if is_datetime64tz_dtype(indices.dtype):
+            # The second.values below will drop tz, so the rest of this test
+            #  is not applicable.
+            return
+
         # GH 10149
         cases = [klass(second.values) for klass in [np.array, Series, list]]
         for case in cases:
diff --git a/pandas/tests/indexes/conftest.py b/pandas/tests/indexes/conftest.py
index b1dcf0ed9b44b..8e0366138f71e 100644
--- a/pandas/tests/indexes/conftest.py
+++ b/pandas/tests/indexes/conftest.py
@@ -7,6 +7,7 @@
     "unicode": tm.makeUnicodeIndex(100),
     "string": tm.makeStringIndex(100),
     "datetime": tm.makeDateIndex(100),
+    "datetime-tz": tm.makeDateIndex(100, tz="US/Pacific"),
     "period": tm.makePeriodIndex(100),
     "timedelta": tm.makeTimedeltaIndex(100),
     "int": tm.makeIntIndex(100),
diff --git a/pandas/tests/indexes/test_numpy_compat.py b/pandas/tests/indexes/test_numpy_compat.py
index 583556656ac87..7051f2b02fe03 100644
--- a/pandas/tests/indexes/test_numpy_compat.py
+++ b/pandas/tests/indexes/test_numpy_compat.py
@@ -81,6 +81,9 @@ def test_numpy_ufuncs_other(indices, func):
 
     idx = indices
     if isinstance(idx, (DatetimeIndex, TimedeltaIndex)):
+        if isinstance(idx, DatetimeIndex) and idx.tz is not None:
+            if func in [np.isfinite, np.isnan, np.isinf]:
+                pytest.xfail(reason="__array_ufunc__ is not defined")
 
         if not _np_version_under1p18 and func in [np.isfinite, np.isinf, np.isnan]:
             # numpy 1.18(dev) changed isinf and isnan to not raise on dt64/tfd64