Ignore burden voltage for the moment.
When placed in a current range, every meter will have a fixed resistance between the terminals. This resistance will depend on the range setting, being very low on the 10 A range and progressively greater on mA and uA ranges. You can measure this resistance using another meter on the ohms setting and therefore find out how much this series resistance is likely to affect your circuit if you insert the meter to measure current.
Secondly, when on the 10 A range you may find that the probe leads themselves add significant resistance to the circuit, especially if they are low quality leads.
Normally speaking, if an inserted DMM on the 10 A range stops your circuit from working, your circuit has a problem.