Above the barrier voltage does a pn junction behave ohmically?
Real diodes have a resistive component to the behavior, not described by the Shockley equation. 

Of course if you have 1.5V (forward bias) across a silicon diode at room temperature (or higher) it is conducting a lot of current. At very high currents (typically well above the rated continuous current) the resistance can dominate. 

Here is a SPICE model for a 1N4004 from Onsemi:

.MODEL D1n4004rl d
+IS=5.31656e-08 RS=0.0392384 N=2 EG=0.6
+XTI=0.05 BV=400 IBV=5e-08 CJO=1e-11
+VJ=0.7 M=0.5 FC=0.5 TT=1e-09
+KF=0 AF=1

You can see there is a series resistance RS in the model of about 39m ohm. If you pulse 10A through this 1A diode, the resistance will drop 392mV.