When set for voltage measurements a multimeter will normally have an impedance measured in megaohms. Although that impedance combined with capacitance may affect the operation of some circuits I can't think of many practical situations where damage is likely to occur. When set for current measurement it's a different story because the shunt resistor may effectively present a short-circuit.
By far the most likely cause of damage (which I've done myself on the odd occasion) are the probes slipping and shorting something out. Apart from being careful in general putting some heatshrink over most of the probe to leave only the tips exposed may help. Also consider getting a set of appropriate test clips for what you're working on so they are physically secure.
Also consider personal safety if you're working at high voltages and make sure any probes you use have an appropriate voltage rating and reduce the chance of your hands slipping onto dangerous voltages.