Protein search for specific sequences on DNA marks the beginning of major biological processes. Experiments indicate that proteins find and recognize their targets quickly and efficiently. Because of the large number of experimental and theoretical investigations, there is a reasonable understanding of the protein search processes in purified in vitro systems. However, the situation is much more complex in live cells where multiple biochemical and biophysical processes can interfere with the protein search dynamics. In this study, we develop a theoretical method that explores the effect of crowding on DNA chains during the protein search. More specifically, the role of static and dynamic obstacles is investigated. The method employs a discrete-state stochastic framework that accounts for most relevant physical and chemical processes in the system. Our approach also provides an analytical description for all dynamic properties. It is found that the presence of the obstacles can significantly modify the protein search dynamics. This effect depends on the size of the obstacles, on the spatial positions of the target and the obstacles, on the nature of the search regime, and on the dynamic nature of the obstacles. It is argued that the crowding on DNA can accelerate or slow down the protein search dynamics depending on these factors. A comparison with existing experimental and theoretical results is presented. Theoretical results are discussed using simple physical-chemical arguments, and they are also tested with extensive Monte Carlo computer simulations.