Will genetically modified food affect our health?
It's a popular public sentiment that - GM foods like tomatoes (flavr
  savr) will affect our health.. Is there any logical scientific
  explanation behind this?


GM organisms can have different genes and gene expression than natural ones, so they can create more protein or different proteins. Proteins can be toxic, can cause allergy and can catalyze reactions which can lead to toxic products.

To create a GM plant you need dollar millions, so it is not a cheap business. The GMO food regulations are very strict (at least in the EU), so it is unlikely that you will eat a GMO which contains toxic compounds.


  Whereas food safety organizations find nothing inherently unsafe in
  the process of genetic engineering, they note that new plant
  characteristics developed in the next wave of agricultural
  biotechnology could present new risks. At the same time, future
  generations of biotechnology may well provide food products with
  higher vitamin or protein content and lower allergenicity, yielding
  benefits to food consumers.



2014 - Agricultural Biotechnology: The Promise and Prospects of Genetically Modified Crops


Despite all the effort to make GMO foods safe, there are issues:


  Pesticides associated to genetically modified foods (PAGMF), are
  engineered to tolerate herbicides such as glyphosate (GLYP) and
  gluphosinate (GLUF) or insecticides such as the bacterial toxin
  bacillus thuringiensis (Bt). The aim of this study was to evaluate the
  correlation between maternal and fetal exposure, and to determine
  exposure levels of GLYP and its metabolite aminomethyl phosphoric acid
  (AMPA), GLUF and its metabolite 3-methylphosphinicopropionic acid
  (3-MPPA) and Cry1Ab protein (a Bt toxin) in Eastern Townships of
  Quebec, Canada. Blood of thirty pregnant women (PW) and thirty-nine
  nonpregnant women (NPW) were studied. Serum GLYP and GLUF were
  detected in NPW and not detected in PW. Serum 3-MPPA and CryAb1 toxin
  were detected in PW, their fetuses and NPW. This is the first study to
  reveal the presence of circulating PAGMF in women with and without
  pregnancy, paving the way for a new field in reproductive toxicology
  including nutrition and utero-placental toxicities.



2011 - Maternal and fetal exposure to pesticides associated to genetically modified foods in Eastern Townships of Quebec, Canada



  The study of combined effects of pesticides represents a challenge for
  toxicology. In the case of the new growing generation of genetically
  modified (GM) plants with stacked traits, glyphosate-based herbicides
  (like Roundup) residues are present in the Roundup-tolerant edible
  plants (especially corns) and mixed with modified Bt insecticidal
  toxins that are produced by the GM plants themselves. The potential
  side effects of these combined pesticides on human cells are
  investigated in this work. Here we have tested for the very first time
  Cry1Ab and Cry1Ac Bt toxins (10?ppb to 100?ppm) on the human embryonic
  kidney cell line 293, as well as their combined actions with Roundup,
  within 24?h, on three biomarkers of cell death: measurements of
  mitochondrial succinate dehydrogenase, adenylate kinase release by
  membrane alterations and caspase 3/7 inductions. Cry1Ab caused cell
  death from 100?ppm. For Cry1Ac, under such conditions, no effects were
  detected. The Roundup tested alone from 1 to 20 000?ppm is necrotic
  and apoptotic from 50?ppm, far below agricultural dilutions (50%
  lethal concentration 57.5?ppm). The only measured significant combined
  effect was that Cry1Ab and Cry1Ac reduced caspases 3/7 activations
  induced by Roundup; this could delay the activation of apoptosis.
  There was the same tendency for the other markers. In these results,
  we argue that modified Bt toxins are not inert on nontarget human
  cells, and that they can present combined side-effects with other
  residues of pesticides specific to GM plants.



2012 - Cytotoxicity on human cells of Cry1Ab and Cry1Ac Bt insecticidal toxins alone or with a glyphosate-based herbicide



  Though GM crops promises to meet the world's food need and eliminate
  the world hunger and starvation, the potent harmful effects cannot be
  overlooked and these need to be taken seriously. The long term impact
  on the society cannot be overruled as it includes the health risk of
  all the people involved be it producers (farmers) or the consumers
  along with the scientists who are working in the laboratory. Hence, it
  is the need of the hour to reach and try for sustainable development
  that meets the needs of humans and the balance of nature maintained.



2013 - Cry1Ab Treatment Has No Effects on Viability of Cultured Porcine Intestinal Cells, but Triggers Hsp70 Expression


I think it is the same story as by food, pharma and plastic industry. The billion dollar profit can be more important than the consequences. Btw. if you check a regular food, which you can buy in the shop, you will find a lot of unhealthy components e.g. trans fatty acids, omega-6 fatty acids, fungicide remnants, etc... which can affect your health as well.

My conclusion is that in most of the cases it is safe, however in rare cases it might have a mild harmful effect. To filter out those cases and their causes we need further studies and better regulation.