diff --git a/doc/source/whatsnew/v1.0.0.rst b/doc/source/whatsnew/v1.0.0.rst
index c072bfeff4a72..c4fca8eba784d 100644
--- a/doc/source/whatsnew/v1.0.0.rst
+++ b/doc/source/whatsnew/v1.0.0.rst
@@ -711,7 +711,7 @@ Datetimelike
 - Bug in :func:`pandas.to_datetime` when called with ``None`` raising ``TypeError`` instead of returning ``NaT`` (:issue:`30011`)
 - Bug in :func:`pandas.to_datetime` failing for `deques` when using ``cache=True`` (the default) (:issue:`29403`)
 - Bug in :meth:`Series.item` with ``datetime64`` or ``timedelta64`` dtype, :meth:`DatetimeIndex.item`, and :meth:`TimedeltaIndex.item` returning an integer instead of a :class:`Timestamp` or :class:`Timedelta` (:issue:`30175`)
--
+- Bug in :class:`DatetimeIndex` addition when adding a non-optimized :class:`DateOffset` incorrectly dropping timezone information (:issue:`30336`)
 
 Timedelta
 ^^^^^^^^^
diff --git a/pandas/core/arrays/datetimes.py b/pandas/core/arrays/datetimes.py
index e41f2a840d151..10669b09cefec 100644
--- a/pandas/core/arrays/datetimes.py
+++ b/pandas/core/arrays/datetimes.py
@@ -794,9 +794,7 @@ def _add_offset(self, offset):
                 values = self.tz_localize(None)
             else:
                 values = self
-            result = offset.apply_index(values)
-            if self.tz is not None:
-                result = result.tz_localize(self.tz)
+            result = offset.apply_index(values).tz_localize(self.tz)
 
         except NotImplementedError:
             warnings.warn(
@@ -804,6 +802,9 @@ def _add_offset(self, offset):
                 PerformanceWarning,
             )
             result = self.astype("O") + offset
+            if len(self) == 0:
+                # _from_sequence won't be able to infer self.tz
+                return type(self)._from_sequence(result).tz_localize(self.tz)
 
         return type(self)._from_sequence(result, freq="infer")
 
diff --git a/pandas/tests/tseries/offsets/test_offsets.py b/pandas/tests/tseries/offsets/test_offsets.py
index 458d69c1d3216..6f628bf86829a 100644
--- a/pandas/tests/tseries/offsets/test_offsets.py
+++ b/pandas/tests/tseries/offsets/test_offsets.py
@@ -20,6 +20,7 @@
 from pandas._libs.tslibs.offsets import ApplyTypeError
 import pandas.compat as compat
 from pandas.compat.numpy import np_datetime64_compat
+from pandas.errors import PerformanceWarning
 
 from pandas.core.indexes.datetimes import DatetimeIndex, _to_M8, date_range
 from pandas.core.series import Series
@@ -43,7 +44,10 @@
     CBMonthBegin,
     CBMonthEnd,
     CDay,
+    CustomBusinessDay,
     CustomBusinessHour,
+    CustomBusinessMonthBegin,
+    CustomBusinessMonthEnd,
     DateOffset,
     Day,
     Easter,
@@ -607,6 +611,46 @@ def test_add(self, offset_types, tz_naive_fixture):
         assert isinstance(result, Timestamp)
         assert result == expected_localize
 
+    def test_add_empty_datetimeindex(self, offset_types, tz_naive_fixture):
+        # GH#12724, GH#30336
+        offset_s = self._get_offset(offset_types)
+
+        dti = DatetimeIndex([], tz=tz_naive_fixture)
+
+        warn = None
+        if isinstance(
+            offset_s,
+            (
+                Easter,
+                WeekOfMonth,
+                LastWeekOfMonth,
+                CustomBusinessDay,
+                BusinessHour,
+                CustomBusinessHour,
+                CustomBusinessMonthBegin,
+                CustomBusinessMonthEnd,
+                FY5253,
+                FY5253Quarter,
+            ),
+        ):
+            # We don't have an optimized apply_index
+            warn = PerformanceWarning
+
+        with tm.assert_produces_warning(warn):
+            result = dti + offset_s
+        tm.assert_index_equal(result, dti)
+        with tm.assert_produces_warning(warn):
+            result = offset_s + dti
+        tm.assert_index_equal(result, dti)
+
+        dta = dti._data
+        with tm.assert_produces_warning(warn):
+            result = dta + offset_s
+        tm.assert_equal(result, dta)
+        with tm.assert_produces_warning(warn):
+            result = offset_s + dta
+        tm.assert_equal(result, dta)
+
     def test_pickle_v0_15_2(self, datapath):
         offsets = {
             "DateOffset": DateOffset(years=1),