If two objects of different temperatures have the same heat source applied, do they heat up by the same amount? I have two objects, A and B. A and B are identical in every way except their temperature. Units are not relevant: assume degrees C, F, or anything linear with them.
Also not relevant are phase transitions. Assume that either nothing is freezing/melting/otherwise, or at least that the effects are negligible. The matter is mostly or entirely in the same phase throughout the event.
The temperature of A is 100 degrees.
The temperature of B is 200 degrees.
Applying a heat source raises the temperature of A by 10 degrees up to 110 degrees.
With no modifications, the heat source is applied to B. Again, all other things being equal: should B be expected to heat up by 10 degrees also, or by some other amount?
For an example, consider a room that you want to warm with a fire. 1) The temperature of the room is 10 degrees and the addition of the fire warms it by 20 degrees up to 30 degrees. OR 2) The temperature of the room is actually -50 degrees, but will that same fire warm the room by 20 degrees up to -30?

I ask this because I thought the temperature change would be different in the two different situations. Taking the above example, I would have thought the -50 degree room would be warmed by more than 20 degrees (25? 30? I don't know, just more than 20.) But someone is telling me that the -50 degree room would be warmed by the same 20 degrees, not more.
My thought process is along these lines: Two objects will attempt to equalize their heat when brought together, so 10 degree water added to 30 degree water will result in approximately 20 degree water, but add that same water to 50 degree water and it will result in 30 degree water instead (+/- 20 instead of 10). I realize this is not the same as a heat source, but I thought something similar would apply.
 A: It depends if the heat transfer properties of the materials are temperature dependent or not. 
For example, let's think about the heat capacity, which is a measure of how much energy is required to produce a certain change in temperature. If the heat capacity is constant at all temperatures, then for both cases the objects will have the same change in temperature. If, however, the heat capacity decreases with increasing temperature, then the hotter object will not get as much of a temperature increase as the colder object.
All of this also depends on how long the heat source is supplied and the temperature dependency on other things like conductance of the objects. If everything is independent of temperature, and if the objects are identical with regards to their material properties, then yes, the two objects will experience the same temperature change.
A: It depends on how you choose to define "same heat source".
If you are saying that the net heat transfer is the same in either scenario, then I agree with the other answers.  It would all depend on the material properties and if they vary with temperature. 
Your wording and example implies something a bit different to me though.  You seem to be suggesting that we would be using the same heat source, at the same temperature, and then comparing the final temperatures of the heated objects.  In that case; the starting temperature is very important.
This is because the rate of heat transfer is relative to temperature difference.  A 400 degree campfire will raise the temperature of a 100 degree stick more than a 200 degree stick; all other factors the same.  That's because the 100 degree stick will get a higher net energy input from the flame; due to the greater thermal gradient driving the heat transfer. 
A: The answer to this question is strongly material dependent! 
1st law thermodynamics tells us that if we heat up a body  by an infinitesimal amount dQ but with no work done and no volume changes on the body, then dQ=dU, where dU stands for a corresponding infinitesimal increase in the internal energy of the body. Now, we said there are no volume changes, we naively expect the internal energy to depend only on temperature. Therefore we can write:
$dQ=\big(\frac{dU}{dT}\big)dT=mC(T)dT$, $\hspace{0.3cm}$ $mC(T):= \frac{dU}{dT}$
giving us the well known equation that describes how heat translates into  temperature differences, with a small twist! Here the specific heat can depend on the temperature.
Now in most introductory contexts, it is usually assumed that the specific heat $C(T)$ is a temperature independent constant. In some real materials (e.g. metals at some high temperature, gases close to a classical ideal one) assuming the specific heat is a constant is a great approximation and indeed for the same amount of heat you'll get the same increase in temperature no matter what your initial temperature is, because then it is true that:
$Q=mC(T_f-T_i)$
and the temperature difference is directly proportional to the heat distributed.
However if the material's specific heat does depend on temperature strongly, ($C$ is not a constant anymore), then this not true anymore. I'll finish with an example. For a cold gas of free electrons, it is well known that $C(T)=\gamma T$, with $\gamma$ being a constant. By integrating the equation above one can see that for a specific amount of heat Q given the following holds: 
$Q=m\gamma\frac{T_f^2-T_i^2}{2}$
so for different initial temperatures $T_i$ but the same amount of heat $Q$ you can see that you don't get the same temperature differences.
A: 
  
*
  
*The temperature of A is 100 degrees.
  
*The temperature of B is 200 degrees.
  
*Applying a heat source raises the temperature of A by 10 degrees up to 110 degrees.
With no modifications, the heat source is applied to B. Again, all other things being equal: should B be expected to heat up by 10 degrees also, or by some other amount?



*

*Object A starts at 100 degrees.
You apply a temperature of 110 degrees to A.
Eventually object A reaches a temperature of 110 degrees because you said: "Applying a heat source raises the temperature of A by 10 degrees up to 110 degrees." - IE: your heat source does not cool or run out of fuel until object A reaches 110 degrees, at which point the heat source is removed from the equation and object A continues to heat or commences cooling to match the surrounding temperature.

*Object B starts at 200 degrees.
You apply a temperature of 110 degrees to B.
Eventually object B reaches a temperature of 110 degrees because you said: "With no modifications, the heat source is applied to B. Again, all other things being equal: should B be expected to ...".
The source of heat used to heat A is identical to what was used with object B.
Object B loses it's heat because hot flows to cold
A more easy to visualize example might be two identical hot water bottles with equal volumes of warm water, both at the same temperature. One hot water bottle is given to someone whom is freezing cold, possibly just walked out of a walk-in freezer. The other hot water bottle is given to someone whom is on the beach on a very hot day.
Both persons extract all the available heat from the warm water but the freezing person finds that the bottle is hot while the overheated person finds the warm water somewhat cool.
