Holy ground is the only sanctuary open to immortals. According to their rules, no immortal combat may occur on Holy Ground. Whether this means that immortals may not fight or may not kill on holy ground is debatable.
There is some indication that an immortal may kill a passive immortal on holy ground with no negative repercussions, though this is almost entirely speculative (see the Sanctuary). There is a legend / rumor that prior to the eruption of Vesuvius, two immortals were battling in a temple of Apollo. Generally only the most ignorant or deranged immortal would even attempt to engage in combat on holy ground. Even mortals seem to be safe from immortals in the sanctuary of Holy ground as seen when Ernst Daimler waited to attack Father Bernard until after he fled his church (Mortal Sins).
What exactly qualifies as holy ground and how an immortal can know the ground is blessed is never explicitly stated. There are cases of qualifying holy ground including land blessed by non-Christian faiths and clergy but there is no explicit statement as to how two immortals would know of a patch of ground's status unless it were indicated to be holy via markings, etc. It is possible that should the immortals engage in combat there would be immediate indications that the rule was being violated such as earth tremors, etc. such as during the fight between Connor MacLeod and the warrior Kane in 1994.
The exact rules regarding holy ground are difficult to determine. Kiem Sun challenged Ducan MacLeod to a friendly sparing match; the two fought briefly without consequence, suggesting that Immortals can "fight" on holy ground provided they refrain from serious injury. By contrast in the film Highlander: Endgame, Kale murders numerous immortals while they are asleep. The logic offered by the film is that because Kale's victims could not defend themselves, his actions did not qualify as doing battle and so did not violate the rules.
In Kurgan Rising, Duncan is lured into taking another immortal's head on consecrated ground. MacLeod notes that he should have instinctively realized the ground was consecrated implying that the immortals have an ability similar to their Buzz that acts as a warning system. It is later revealed that the building Duncan committed the transgression in had been specifically designed to channel such an act and allow the mad mortal who built it to resurrect the Kurgan implying that something about the structure itself crippled MacLeod's warning system. Note: Kurgan Rising is not considered canon.