Would there be any interest in a manned mission to Mars without landing? This could consists of none (just a flyby) to many revolution(s) around the planet. Would it cut a consequent part of the cost compared to a landing and return mission?
Yes, there is interest, at least from some quarters. The Planetary Society proposes an orbit-first strategy for humans at Mars as part of an affordable plan to make progress and build capability. Yes, putting humans into orbit around Mars and returning from there would be much, much cheaper than landing them and getting them back off the surface.
While there, the crew could teleoperate rovers on Mars far, far more capably than we can do from Earth due to the light-time difference. The rovers themselves would be much more advanced than what we have today, and would be able to explore wider ranging, diverse terrains across the planet, more than the humans on a single landed mission could. The crew can also land on the moons of Mars and explore more directly there, both for science and for resource prospecting.
I see here heartfelt concerns for the astronauts taking all that risk without sufficient reward. You can set your concerns aside. I am confident that you will find many extremely highly qualified candidates who are willing to go on a Mars orbit and Mars moon rendezvous mission.
Do you know the names of the first humans who landed on the Moon? Yes! Armstrong and Aldrin, of course. (and if you scratch your head a little, you remember Collins in orbit too). Do you know the names of the first who orbited the the Moon? (Borman, Lovell and Anders). Less likely. My point is: If you later land on Mars, people forget about the first mission. And if you do not follow up with a landing, people feel like you have cheated.
From a technical point of view, such a mission is just a long stay on a single use space station, without the possibility of a return in case of an emergency. On the other side, the technology is pretty much off-the-shelf. The much lower $\Delta v$ requirements and lower complexity are undoubtely cutting a large part of the cost too.
In short, a moderate risk, high cost voyage, with only a small reward.
I think if a mission was to be sent to Mars without the intention of landing, there could be a great deal of science and hardware development that could be done by landing on Phobos (escape velocity ~41km/hr).
According to one NASA proposal:
Precursor missions to the surface of Phobos prior to undertaking human Martian surface operations would allow for a continuation in the phased approach of development, implementation, and in-situ testing/experience of the necessary hardware for an eventual Mars surface mission.
This plan calls for development of surface habitat/landing vehicle and a rover vehicle which could later be used on future missions to the Martian surface.
Also, the Caltech Space Challenge in 2013 did some interesting development work on a Phobos mission. They devised a plan using 11 Spacex launches to assemble the mission hardware in LEO, with a total mission duration of ~450 days. However this plan does involve a "nuclear propulsion unit" which may need a bit of development work first.
When a crew would arrive in the proximity of Mars (during a flyby) or enter its orbit, they'd be unable to return to Earth because, from a heliocentric point of view, the Earth goes faster around the Sun as it's closer. Therefore, the crew would either also have to go around the Sun and even faster (which requires a lot of fuel) or make a swing-by around Mars and go the opposite direction (which would also require a lot of fuel to decelerate near the Earth). The longer they stay in Martian orbit the more fuel they'd need for returning to Earth. Therefore a Mars flyby would be the better option than orbital insertion.
The first space tourist Dennis Tito founded Inspiration Mars - A Mission for America to make a married couple representing mankind go around Mars in a flyby as described above in the first option: by going on around the Sun eventually "catching up" with Earth again. Inspiration Mars Foundation
Tito's concepts were never considered much by any space agency and the Foundation is seemingly dissolved, their webpage defunct and, unfortunately, it won't be realized by Inspiration Mars.
But it would more reasonable that the first manned flight to Mars also aims to land there. A crew would fly to Mars about six months, stay on Mars an (Earth's) year until the planets are going to "meet" each other again and fly back to Earth another six months. That's risky, and perhaps by nuclear pulse propulsion it would be better to go to Mars in both flybys and landing missions, such a Mars flight would take by far not as long as a conventional one. And a Venus flyby would be easier than one around Mars and might perhaps be carried out first.