Recently, phages are used against the MDR pathogen in humans to combat the diseases. Wouldn’t our immune system try to eliminate those bacteriophages? How these techniques are successful?
Phages have been used in human therapy for many many years, so this is not a new approach. As to how they escape our immune system, first it will depend upon the mode of delivery, for example if administered orally it won't really interact in any significant way with our immune system, so that should be fine. It might be more of a problem were the phages to be delivered by injection into the circulatory system. But more relevant is that the phages are likely to have acted very rapidly (hours) whereas an immune response will take days.
It may well be true that for some bacteriophages they would only be effective one time if the body did mount an immune response, but again this would primarily only be relevant for phage in the circulatory system.
Aravindharajan S.T.M. Phages were used with success for nearly 100 years.
Kuchment A. The Forgotten Cure: The Past and Future of Phage Therapy. Springer Science & Business Media; 2011 Dec 9.
They are often applied with/after antibiotics treatment even for newborn kids and elderly persons. In our family, we always had some fresh bacteriophages against Staphylococcus/Streptococcus/toxic E.coli infection in our fridge when kids were small (0-7 y.o.). Sure, the exact phage cocktail must be prescribed after bacteria analysis only.
If I understand correctly you are asking how phage therapy deals with the potential for the phage to facilitate virulence/pathogenic genes being exchanged within the microbial flora and leading to a drug resistant or otherwise enhanced pathogen that survives.
I believe one way is by engineering or selecting phage that are only lytic so that there is no chance for phage to recombine with the bacteria and pick up pathogenic genetic elements to pass on. It is also usually more desirable in phage therapy since the goal is killing the bacteria. I have seen papers that discuss using alternative lysogenic phage but that is not the norm.
As Binyamin Kerman mentions, it is possible to exchange some genes between phage and host by recombination. However the level of recombination will be vastly reduced with a lyric phage. More appropriately though is I would hope any phage used in therapy would have been sequenced so that it is known if it carries any genes that are known to promote host virulence.