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Some websites recommend a pair of load resistors for LED turn signal lights. Will it work or burn out faster if I don't get load resistors?
Very Easy.
Original lamp is 3157NAK
3157 NA K Data
This lamp has 2.1 Amp or 27 Watts nominal.
So its internal resistor is:
R = V / I = 13'8 / 2.1 = 6.57 Ohm

You don't need to use so low value. For example 10 Ohm is enough.
With 10 Ohm you have a power value:
W = V x I = V x V /R = 13.8 x 13.8 / 10 = 19 Wats
You need more than 19 Wats. Solution is a metallic resistor that you have to screew on a metallic car part with more than 19 Watts,

For example: 10 Ohm / 50Wat
PDF Data Resistor

You can find it easy at electronic component stores or eBay
10 Ohm 1% 50 Watt Aluminum Resistor
 
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