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If you MOS's gate by MOS's base, then the current will be zero because the gate is a poly then an insulator .This insulator (usually SiO2) doesn't allow DC current to go through the gate. For the diode connected transistor (drain and gate shorted), the drain of one transistor is connected only to gates of 2 transistors and current requires a closed path to circulate. If we say that current can come from the drain side, it won't find another place to flow as both gates can't let current pass, so there'll be no closed path and thus current won't circulate and the current to the gates will be zero.
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