I'm currently a Master's student in my final year and I want to start a Ph.D. next year. I really like doing research (one of my papers got published in a fairly reputable journal), but I more or less dislike the teaching aspect that comes with the Ph.D. title. Is there any alternative way to do research, outside of industry, and not be obligated to teach?
Yes. There are national laboratories like the US DOE labs (Sandia, Oak Ridge, Los Alamos, etc), non-profit independent labs, industrial labs, and even university labs like the one I work at. You will find that many of these kinds of labs have some sort of service mission instead of teaching, but others are purely research focused. For example, researchers in my group also support users of our supercomputers when they run into problems. There are lots of these non-academic research opportunities out there in the STEM fields. I think there are probably many less in non-technical areas.
It depends a lot on the type of research you'd like to do. For example, suppose you're in mathematics:
There are many jobs that use applied mathematics in highly directed R&D activities, where you are working in a team on some bigger project, which is not under your control unless you really rise in the ranks. Academic publications will not be a primary outcome, but applied research could be a substantial part of the job.
There are a moderate number of more academic-style applied mathematics positions, for example in national labs, where you are in charge of your own activities (subject to securing funding) and write many research papers.
There are only a tiny number of secure, long-term jobs where you can do whatever mathematics you would like, including pure mathematics, with no teaching responsibilities (and where doing mathematics is your primary job).
So the availability of research-focused jobs really depends on how flexible your interests are and how well they fit with other people's goals. If your work is obviously and immediately applicable to industry or government, then there's a good chance you'll be able to convince someone to pay you to do it full time. If you are doing applied work that is less immediately applicable, then it depends on the availability of funding. If you are doing highly theoretical work, then you'll have to be extraordinarily skilled or lucky.
There are many institutes called "Academy of Sciences" all around the world. In general, being member of these, you're not obliged to teach nor to supervise students. You can still be assigned some other responsibility than teaching, but it should not be really limiting you.
As an example, this exists in France as CNRS (Le Centre national de la recherche scientifique) and in Czechia as CAS/ASCR/AVČR (The Czech Academy of Sciences).
As for France, note that getting a CNRS position is very difficult, the positions are literally couple in each branch each year. Most people in CNRS laboratories (institutes) have teaching positions at the associated universities.
I've heard that the Center for Communications Research is like this, if your Ph.D. is in math. I know people who work there and quite like it. (US Government; a security clearance is required.)
In the UK, places like the JIC,TGAC, Sainsbury Lab and IFR on the Norwich Research Park are affiliated with the University of East Anglia but are a significant distance from and have no real connection to the undergraduates therefore no teaching is expected. There are quite a lot of places like this in the UK, like CRUK and MRC as well where you are not expected to teach.
NB: You should look out for institutes funded privately or through charities as these are the ones that are more independent and less likely to have teaching commitments, if at all.