Most places have a pit area that competitors set up and roll up theirs lines when done with a heat.  Here when we had a circle we were allowed to roll out lines at start of a heat and roll them up when done.  But when we had a lot of entries years ago we a pit/ready area.  Also pull test and measurement was done on the circle.

It is up to the pilot that is still flying, that when he runs out of fuel or shuts off to make sure he does get tangled with the lines on the ground of the plane or planes that are down.  This especially true if planes are making pit stops as the line may not be on the ground if the plane is being pitted.   Same with taking off you/pilots must make sure to fly over the lines if the plane of competitor has not gotten air born yet.   If some one is pitting and you come in for a stop/landing you must make sure you don't snag his/her lines.  But if the person on the ground picks up his/her plane and causes you to snag then he/she is guilty of causing a snag.  I have also seen teams DQ'd because they launched while a p[lane was coming in.   

It's all spelled out in the AMA rule book, so take the time to read the rules.

Start out with the CL Racing unified rules.

Look at and understand section 7. That section describes the field lay out, i.e., where all of the circles are painted and what they are used for. as an example,the "Outer Circle" is the innermost limit for pitting operations. And other words for the other circles.

Then look at section 8, labeled "Field Requirements" but is more a set of rules governing pilot and pit crew actions during a race.

Many other items appear as well in earlier and later sections, but perhaps to answer your question directly, and to perhaps clear some wrong information,  section 10.6 reads as follows:

10.6.
It is the pit crew’s/pilot’s responsibility to keep the control lines on the
ground during a start, pit stop, or after completing a race. Should failure to
comply with this rule lead to the snagging of a second aircraft which is
taking off or landing, and if, in the Contest Director’s opinion, the snag
affected the race results of the second aircraft, the entrant whose lines
were snagged shall be disqualified, and the entrant who snagged the lines
will have his race rescheduled for a later time. If, in the Contest Director’s
opinion, the line snag was not due to negligence, and the snag affected
either entrant’s race results, either or both entrants may be rescheduled for
a later race and no disqualification need result.

In other words, the landing model has priority and CAN ROLL THROUGH YOUR LINES. (This in direct opposition to the FAI F2C rules.) So you keep the lines on the ground when a model is landing over you.