Does it cost more to run a refrigerator on an extension cord?
Yes, it does cost you money using the extension cord.  

Grabbing a random number out of the air, let's say the fridge draws 2 amperes. If you have a 50 ft. 16 AWG cord, that's 1.996 watts. If the cord was 50 ft. 14 AWG, that'd be 1.256 watts.

Using the energy star standard of $0.12 per KWh. The 16 AWG cord costs you $0.00023952 per hour, while the 14 AWG cord would cost $0.00015072 per hour.

There's also the problem that the manufacturer's documentation likely tells you not to use an extension cord.

NOTES:
- I have no idea how much current a mini fridge draws, so I simply grabbed a number out of the air. If somebody wants to comment with a realistic number, I'll rerun the calculations. 



Using @CarlWitthoft's calculation of 0.7 amperes, and resistance values from NEC chapter 9 table 8. The cost can be figured as follows.

18 AWG cord

VD = Length x 2 x current x ohms per foot
VD = 50' x 2 x 0.7 A x 0.00795 ohms
VD = 100 x 0.7 A x 0.00795 ohms
VD = 70 x 0.00795 ohms
VD = 0.5565 volts

Watts = Volts x Amperes
W = 0.5565 V x 0.7 A
W = 0.38955 watts
kW = 0.00038955  

Cost = kW x $/kW
Cost = 0.00038955 x $0.12
Cost = $0.000046746

16 AWG cord

VD = 70 x 0.00499 ohms
VD = 0.3493 volts  

W = 0.3493 V x 0.7 A
kW = 0.00024451  

Cost = 0.0002445 kW x $0.12
Cost = $0.0000293412

14 AWG cord

VD = 70 x 0.00314 ohms
VD = 0.2198 volts  

W = 0.2198 V x 0.7 A
kW = 0.00015386
Cost = 0.00015386 kW x $0.12
Cost = $0.0000184632

So for every hour the fridge is actually running, you'll pay the above amount extra because of the cord.



If you're using the 18 AWG cord, and the fridge runs constantly for a year. The power lost to the cord would only cost about $0.40. 

$0.000046746 x 24 hours = $0.001121904
$0.001121904 x 365 days = $0.40949496

So basically, reading this ridiculous answer has likely cost you more than the cord will.