I presume by "air intake" that you are talking about the turbine air inlet filter system.
Put on a surgical or dust mask covering your mouth and nose. Now breathe through that mask. How does it affect your breathing? As the mask gets dirtier, how does it affect your ability to breathe?
It's the same for any internal combustion engine that uses a filter to prevent dirt and dust from entering the engine--as the filter gets dirty the ability to draw air into the engine is reduced, so the efficiency of the machine is reduced. It has to work harder to get less air into the engine, and since most engines (including combustion ("gas") turbines) make more power when the air flow is high, they can't make as much power when the air entering the engine is low or lower.
If by "air intake system" you mean something other than turbine air inlet filter, then you need to be more specific.
There is just too much variation between turbines and sites and ambient conditions to develop a general rule of thumb--other than to say that increasing filter dp increases heat rate.
Something to be aware of is that as the filter dp increases the forces on the inlet duct panels increases. There have been many sites that delayed changing filters based on some spreadsheet analysis that said the decrease in performance didn't justify changing the filters for some period of time. At some of these sites, the filter elements ruptured and dirt was ingested into the compressor. At other sites the inlet duct work collapsed and in a couple of cases metal pieces were ingested into the compressor causing catastrophic FOD (foreign object damage) which took hundreds of thousands of US dollars to repair, along with the loss of hundreds of thousands of US dollars in generation and steam production (sometimes steam is the most important product!).
The type of ambient contaminant, the ambient humidity, the site elevation, the compressor condition and cleanliness can all have a significant impact on heat rate. How does one separate these factors from filter dp when analyzing heat rate factors and unit performance?
Optimizing heat rate is a function of many things, and ambient conditions, as well as how the unit is being operated (Part Load, Base Load, IGV Temp Control On or -Off, etc.) all have a very large impact, too.
From a Design Engineering point of view, the rule of thumb is that Gas Turbine Output Power drops by a factor of 0.004 per inch water of inlet pressure drop and a factor of 0.00167 per inch water of exhaust pressure loss. On the other hand the Heat Rate increases by a factor of 0.0167 per inch water of inlet pressure drop and a factor of 0.00167 per inch water of exhaust pressure loss. Consult the Gas Processors and Suppliers Association Engineering (GPSA) Data Handbook for detail.