This is receiving a lot of down-votes because people are afraid of the subject matter appealing to Holocaust Deniers who want to argue there was no planned mass execution.
I understand the worries over this, but it is a valid question, with an answer in the affirmative: Yes, large numbers of victims of the Holocaust/Nazi policies died of disease.
Firstly, we have to understand that the Holocaust, which was the deliberate elimination of "undesireables" was not solely the use of crematoriums and gas chambers, but a systematic method of elimination of the Jews and other "undesirables" by the Nazi regime. The deliberate construction of Death Camps, which were different than the Labor Camps, did not begin in earnest until in 1942 after the Wansee Confrence, where Nazi leadership under Reynhard Heydrich and Adolph Eichmann outlined the plans for what we know know as Auschwitz, Sobibor, and Treblinka.
The first Nazi camps were started as early as 1933, to hold political opponents, homosexuals, gypsies, etc. These existed as labor camps or simply large prisons. In these camps, they largely served as slave labor for war industries, often in munitions factories, mines, etc. However, due to crowded conditions and lack of proper care, thousands died at these camps alone. By 1945, at Dachau over 200 people a day were dying from poor treatment alone.
Socially communicable diseases, almost all caused by poor sanitation and hygine, such as Typhus, were rampant in these camps. With Dachau alone holding an estimated 200,000 people in just the capacity as forced laborers, over 30,000 are known to have died there from malnutrition, over working, disease, and yes, outright murder by camp guards for their own amusement.
At Mauthausen a mining labor camp in Austria, over 3,000 inmates died of hypothermia after they were forced to take cold showers and then stand exposed outside in freezing conditions. At the same camp Karl Gross, a Nazi doctor deliberately injected diseases into inmates to study the effects of the disease. About 1,500 are believed to have died this way alone. At this camp there were the famous "Stairs of Death", where prisoners were forced to carry large blocks of stone up the side of the quarry, leading many to collapse from exhaustion causing those behind them to stumble and fall creating a sort of domino effect, crushing many to death.
Of course, as many people as these camps could hold, they still never had enough capacity. It was not until the Warsaw Ghetto uprising in 1943, that the Ghetto was finally emptied. At it's peak it held over 400,000 people in an area that took up less that 3% of the entire Warsaw metropolitan area. It's estimated that over 100,000 died of disease there alone.
The conditions of cramming so many people into one area with minimal health care and sanitary conditions were rife with diseases as varied as Typhus, Malaria, Dysentery, combined with food rations that often were less than 1000 calories a day let to even cases of scurvy occurring. At Jasenovac camp, open pit latrines which are gigantic incubators of disease often overflowed due to rains which would recede back into the nearby lake and river which is where the prisoners drew their untreated drinking water.
Now mind you, many people when they want to deny the holocaust point out that "There's no way the crematoriums and/or gas chambers could kill that many people!" They forget that literally hundreds of thousands died by disease alone in the Ghettos, on the trains transporting them, being worked to death, or simply shot. Until 1942 with the Wansee Confrence, the standard practice of "cleansing" an area was to simply shoot them all. It was common to make the victims dig their own graves before they shot them.. This was the standard practice until the construction of deliberate Death Camps such as Sobibor.
People tend to forget that the Holocaust was not just gas chambers and ovens, but a systematic method of maltreatment, cruelty, slavery, and murder. Gas chambers or no, murder through neglect, maltreatment, starvation, and deliberate torture is still murder.
The Holocaust is not just about the gas chambers, it is about the systematic and terrifyingly efficient way in which unspeakable cruelties and murder were visited upon people for simply being different.
EDIT To give you a frame of reference about disease, of the estimated 620,000 deaths of soldiers in both the Confederate and Union armies in the American Civil War, an estimated 2/3 of them were from disease, not combat.
Depends on what you mean by 'significant'. It definitely formed a non-negligible number of deaths. But there's a lot of doubt about exact numbers of deaths, so an exact percentage isn't possible. If exactly how many were killed is subject to debate (numbers range from 5.1 million according to Hilberg, to nearly 6 millon according to Dawidowicz), determining exactly how all of them died is an even more daunting task. Lots of killings were poorly documented, particularly early on, when killing squads shot people en masse. And not all existing documentation survived the war.
Probably the earliest estimate of this is from Eichmann, in Höttl's testimony for the Nuremburg trials. Höttl quoted Eichmann as saying that up to the time of the conversation (August 1944) that 6 million Jews had been killed, of whom 4 million were killed at death camps, and most of the remaining 2 million had been killed in mass shootings. So according to Eichmann, at most 1/6 of the Jews killed were killed from disease and starvation.
However, judging by how round they are, those numbers can really only be seen as estimates. A number of books have been written estimating the number killed, but there's a lot of doubt. The only attempt I know of is Hilberg, who estimates that 800,000 were killed in Ghettos or camps as a result of starvation and disease. But again, this number can really only be taken as an estimate--there's no reliable way of knowing whether someone was killed outright or died of disease in many cases.
So the conclusion is that coming up with a number is very very challenging, well beyond what can be explored in a reddit post, even on /r/askhistorians. But a rough answer is that the number killed through methods other than direct murder was probably between 10-20% of the Jews killed.
Hopefully /u/estherke will be along with more estimates, and perhaps an answer for the extent of this for the Roma.
A better way of putting it would be privation: starvation, and disease caused by imprisonment in appallingly overcrowded ghettos and camps without medical attention. Raul Hilberg, the foremost authority on the Jewish Holocaust, who is famously conservative in his numbers, has calculated the following in The Destruction of the European Jews:
Died through ghettoisation and general privation in Eastern Europe: over 800,000
Open-air shootings, mainly by the Einsatzgruppen: over 1,300,000
Gassed upon arrival in
Died (gassed, shot, starvation, disease) in Auschwitz after being registered for forced labour: 200,000
Died in non-death camps (shot, disease, starvation): 300,000
Total: 5,100,000
Personally I feel these numbers are too low (the numbers for Treblinka and Auschwitz are generally accepted to be somewhat higher; Hilberg only counts 150,000 dead for all the concentration camps within Germany proper combined, etc), but they may give you an idea of the rough percentages according to cause.
To me personally it does not matter whether a person died by gassing or from hunger or from disease concomitant on the conditions in which they were imprisoned, but you asked, so there you have the numbers.
