Had similar problem on my 87 260E. The clue that I am going by is that it turns on fine when cold and difficult when hot. I am assuming that there is no fuel smell or fuel leak anywhere. Also it is not as trivial as fuel filter. Two parts may be involved. There is a fuel injector temperature sensor that tells the fuel computer how much fuel to send to the engine. It sits on the engine and is subject to very high temperatures, so chances are very high that it has gone bad and so when the car gets hot, enough fuel is not sent. It is a $40 part, I would change this the first. This fixed my car. If it still does not work, have a MB mechanic check the Fuel relay when the car is hot, the fuel relay may be bad and therefore your fuel pump is not working when the car gets hot. This is about a $100 part. Then the parts like fuel pump, fuel regulator and fuel computer get substantially more expensive, so they should be replaced only if you are very sure, for these items, you will have to get a perfect diagnosis, perhaps from a very good and patient MB tech.
Also as an immediate rememdy, in my case if I depress the gas pedel completely, the car would start right away. This may work for you too.
Saumil S. Patel