My name is XXXXX XXXXX X will assist you. Since your truck is a 2006 it cannot have the 7.3 engine, it must be a 6.0. This is a pretty common issue with them. what you are experiencing is what is called a high pressure oil leak. the engine requires 600 psi of high pressure oil in order to start and it cannot acheive this when hot due to a leak. If you test for codes most likely you will find a code P2291 stored. there are 3 main places that they leak. If you remove the valve covers there is a large cast manifold that is bolted over the injectors. this is the oil manifold. towards the front of each manifold there is a plug that will either require a 10mm allen socket to remove it or a 1/2" rachet extension to remove it. when you take them out there are 2 o-rings on the plug, usually the bottom one will deteriorate and seals good enough to start cold, but leaks too badly to seal when oil is hot.
The other place is the "STC" fitting on the back of the high pressure pump. it is a snap together quick disconnect fitting that connects the HPOP to the branch tube assembly. the fitting wears out and leaks oil. There is a updated repair kit that no longer uses a snap together style fitting and eliminates this from happening again. In order to access the pump and fitting the turbo, intake manifold and oil filter adapter and secondary fuel filter will need to be removed.
If your truck is less than 5 years from the original date of sale and has less than 100,000 miles on it the repair will be covered by your diesel warranty (there is a $100 deductable).
I am on vacation this week, any other questions, feel free to ask, but I wont be back on untill sometime later tonight.