Generally yes, but not always.
If you include the APU as an engine, the answer would need to be different, but I understand your question concerns just the engines used for propulsion.
For jets, the direction of rotation doesn't make much of a difference, so the same engines can be used on all stations.
However, with propellers the swirl does influence flying characteristics, especially at low speed, so special left- and right turning versions of turboprop and piston engines are available.
Maintenance and logistics become much simpler if only one type of engine is used.
In the past, not so much emphasis was placed on this, so some airplanes used different types of engines.
Examples were: Convair B-36B , which combined piston and jet engines in the same airframe Junkers G-38 , which used a Junkers L88 inboard and a Junkers L8 outboard.
Later, the outboard engines were also changed to the L88.
This can be seen from the number of propeller blades: If the outer engines drive a two-bladed propeller, it is the earlier version with the L8.
Hawker-Siddeley Trident 3 , which was almost a four-engined jet, because it had a small, tail-mounted RB-162 engine in addition to the three tail-mounted RB-163 Spey to provide thrust on takeoff.
Rutan Voyager , which used a bigger front and a smaller rear engine.
aviation.stackexchange.com/users/1961/Peter Kämpf
The place to find the true answer is in the Type Certificate Data Sheet.
This document will tell you exactly what engines can be installed on an aircraft.
For example on the DC-10-10: 3 General Electric CF6-6D, CF6-6D1, CF6-6D1A, CF6-6K or CF6-6K 2 Turbofan Engines.
(CF6-6D and CF6-6K engines may be intermixed in accordance with Appendix XXII of the applicable FAA Approved Airplane Flight Manual.
CF6-6D1 and CF6-6D1A engines may be intermixed in accordance with page 2.1 of Section IVB of applicable FAA Approved Airplane Flight Manual.)
aviation.stackexchange.com/users/2341/Sports Racer
As this image shows the Navy flew a plane with two propeller engines and two jet engines.
Read more about it here.
From wikipedia:
Beginning with the P2V-5F model, the Neptune became one of the first aircraft in operational service to be fitted with both piston and jet engines.
The Convair B-36, several Boeing C-97 Stratofreighter, Fairchild C-123 Provider, and Avro Shackleton aircraft were also so equipped.
To save the weight and complexity of two separate fuel systems, the jet engines on the P2Vs did not burn jet fuel-
they burned the same fuel as the piston engines: 115–145 Avgas.
The jet pods were fitted with intake doors that were kept closed when the J-34s were not running to prevent them from windmilling, allowing for economical piston-engine-only long-endurance search and patrol operations.
In normal U.S. Navy operations, the jet engines were run at full power (97%) to expedite and assure all takeoffs, then shut down when the aircraft reached a safe altitude.
Also, the jets were started and kept running at flight idle during low-altitude (500 feet during the day and 1,000 feet at night) anti-submarine and/or anti-shipping operations at sea as a safety measure in case one of the radials developed problems.
aviation.stackexchange.com/users/8210/Scooter
Usually the Engines on the same plane are of the same model (or type).
As mentioned in an answer earlier the rotation direction of jet turbines doesn't affect handling, however, the engines installed on either sides must have same thrust as to make the handling easier.
One example for differing engines however is A380, where the inboard engines are equipped with thrust reverser and outboard engines don't have them.
There are several reasons for doing this, not having thrust reverser makes engine comparatively simpler to build and maintain.
aviation.stackexchange.com/users/7863/Mellon En Eina
The "Super 27" replaced the outer engines of the Boeing 727.
The problem with re-engining a trijet is the necessity of re-engineering the center engine's casing and intake, wspecially if an S-duct is involved.
That is so complicated that re-engine plans typically left that engine alone .
So it was with the Super 27 with a swap to the more powerful JT8D-217.
I have jeard of schemes to put even more modern engines on the outer pylons, but cannot find the link.
aviation.stackexchange.com/users/22864/Harper - Reinstate Monica
In general, are there any differences between the engines on a multi-engine aircraft? Are engines designed to be specifically for the left or right wing, or inboard or outboard for a 4-engine aircraft?  What about the tail engine on a tri-jet like the DC-10? Is it the same as the two wing-mounted engines?
Are all the engines on a multi-engine plane the same?
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aviation.stackexchange.com/users/282/Graham Borland
Most jets have the same engines on all stations as the rotation does not make any difference. However, the A380 has inboard engines equipped with thrust revereser and the outboard engines are not. The place to find such info is Type Certificate Data Sheet. The engines on a multi-engine plane are usually the same. As the direction of rotation doesn't tend to make a difference on jet turbines, the same engines can be used. However, the engines are not always the same. The A380 is one example of a plane with different engines. 
The engines on a multi-engine plane are usually the same, but not always. As the direction of rotation doesn't tend to make a difference on jet turbines, the same engines can be used. The Type Certificate Data Sheet is the best place to look for an answer. The A380 is one example of a plane with different engines.