How does the IS/LM model respond to organic GDP growth when the money supply is held constant? If GDP increases (due to tech change/good investment), interest rates would move up along the LM curve and down along the IS curve, so what happens to the equilibrium? Do we assume GDP cannot grow without money supply growth, because an increase in interest rates along the LM curve would kill investment, bringing GDP back to equilibrium?
We would move upward along the LM curve. However if there is growth because of investment or some technology shock that would cause the IS curve to also shift upward. 

IS: When the growth "shock" you mention happens, there is now more demand for goods (Demand= C+I+G in a closed economy), without any change in interest rates. Since demand shifts, supply goes along with it in IS-LM and GDP increases 

The IS captures the GDP-to-interest rate relationship (along the curve) since this relationship has now changed, we must shift the curve. Specifically we have more GDP now per unit of interest rate compared to before. Since interest rates didn't change in your example, but there is more GDP (caused not by an interest rate increase) the IS curve must represent more GDP for each interest rate, which - if you draw it now - is a shift to the right of the IS.

LM: Represents what happens to interest rates when GDP increases. This relationship has not changed, so we don't have a shift. The reason this relationship has not changed is because the only reason interest rates are increasing now, is because of GDP, something the current LM curve already captures. (If it would be for something else, like money supply change, there would have to be a shift.) 

So there can be growth without money supply increases. However since there is now more demand for money (because there are more things that we can buy) but not more supply of money, then money increases in value, i.e. the "price of money" which in a way is the interest rate must go up. For that reason, part of the growth will be reduced (as investment goes down), but we will still have growth. If money supply also increases exactly enough to accomodate the GDP increase (typically is the case in reality) the interest rate will not increase and the LM would also shift and we do not have the mentioned negative effect.