The most obvious answer to this query is "It depends". Each event has its own rules and most (if not all) events have a proviso about having the rights to make judgement calls about a given event even if it falls outside the scope of the provided rules.
That said, the best approach would be to scrutinize the rules behind some of the most renowned LAN events. Dreamhack's relevant ruling regarding drugs is as such:
1.4 Alcohol and drugs in any form do not belong and are not allowed to be taken onto DreamHack premises. People intoxicated and/or under the influence of drugs/other non-allowed substances will not be allowed on the premises, at any time. People found intoxicated and/or on drugs will be shown off the premises. Smoking is strictly forbidden indoors and is only allowed in designated area outside. Electronic cigarettes is not allowed to use inside of DreamHack.
Nothing in the rules as written directly addresses penalties or other actions levied against players specifically, though this does speak to a firm "no, don't do it" attitude at the event. Furthermore, this does not address paraphernalia, under which a crack pipe, bong, etc. would fall by most peoples' definition. Nothing in the rules as written provides a definite answer with regard to drug paraphernalia, but an educated reading in tandem with the "we reserve the rights to have final say in any situation not addressed by the rules" proviso leads me to believe that the right call is to leave that stuff in the hotel room.
No mention of prescription medication is made anywhere within its rules, and as such it is assumed from a prima facie reading that this rule is intended primarily for the prohibition of "overtly intoxicated" behavior more than any animus towards medication or drugs in general. Much like a lack of mention of prescription medication, no mention of testing or penalties aside from eviction from the event (presumably temporarily) is made in any of the rules as they apply to competitors at Dreamhack's various tournaments.
A cursory reading of the player-specific rules of most LAN tournaments, such as Starladder and others, is roughly similar, and alludes such that the same mindset is likely applicable elsewhere.