Why is the series resistor getting hot in this LED driving circuit?
For a resistor

$$p_R = \dfrac{V^2_R}{R}$$

Generally speaking, it is not a good idea to operate a resistor at its power rating.

But, assuming we do operate at 1/4W, we find:

$$V_{R_{max}} = \sqrt{0.25W \cdot 51 \Omega} = 3.57V$$

So, you really would like the voltage across the resistor to be quite a bit less than \$3.57V\$

Now, the voltage across the resistor is:

$$V_R = 15V - 4 \cdot V_D$$

where \$V_D\$ is the nominal voltage across one of the diodes.  So, if

$$V_D \le \dfrac{15 - 3.57}{4} = 2.86V$$

you'll be at or exceeding the maximum power rating of the resistor which will make it very hot indeed.

If this is the case, then you must increase the resistance or power rating or both.