Is fossil fuel is produced by the process of photosynthesis?

Sort of. Or in other Words: almost.

The answer started 160 million years ago, in the oceans. The Biogenic Theory states that microscopic plankton blanketed these ancient oceans; the plankton captured CO2 (carbon dioxide) and used photosynthesis to break the chemical bonds between the carbon and oxygen atoms. When those bonds broke energy was released and the plankton used that energy to move around, split & reproduce, and so on. When each plankton died it drifted to the bottom of the ocean and would settle next to other dead plankton along with other animals and plants.

Over time, as the continents shifted, this organic matter sank deeper into the earth and the carbon was literally cooked by the heat and pressure of the earth. Depending on the depth at which this matter was cooked we ended up with coal & shale, liquid oil, or natural gas.

So, photosynthesis doesn't create fossil fuels directly, but (if you accept the Biogenic Theory) we certainly wouldn' have fossil fuel without photosynthesis occurring inside of plankton. This is why fossil fuels are often called a form of "stored solar" energy.