Replacing a logic OR gate with two diodes
Your simulation software appears to be modeling ideal or CMOS gates, with infinite input impedance, and real diodes, with some reverse leakage current.  Consequently, when you don't use the pull-down resistors, pressing a button lets the current leak backwards through a reverse-biased diode and turn another gate on.
In an actual circuit there may be enough leakage from circuit elements to ground so that this would work -- or maybe not.  At any rate, if you're driving CMOS from a switch, it's a good idea to make sure that the voltage is asserted somehow, either with the switches as pullups and pulldown resistors, or with pullup resistors and the switches pulling down.
(Note that you've left the 100\$\mathrm{k\Omega}\$ pull-down resistors off of the path with the closed switch -- that'll bite you as soon as you open it and close one of the others).