The use of the Bond theme. Over time, the theme has (rightfully) come to be saved for the action scenes. But early on, they would break it out for absolutely anything. Bond is arriving from the airport, checking in to his hotel, checking his room for hidden cameras or microphones... and this epic action music is playing in the background the whole time.
The scene you're referring to IS unintentionally hilarious, but there's a reason why they did it. From Russia With Love was almost completely finished, including the score, when someone decided they wanted to re-use the Bond theme from Dr.No to help tie the two movies together. The only place in the movie they could find to use it (that wasn't already scored) was the hotel room scene. With the loud, blaring, driving theme playing in the background, it makes the search for bugs and cameras seem like the most exciting part of international espionage!
"The only place in the movie they could find to use it (that wasn't already scored) was the hotel room scene."
This statement seems a bit strange, when you realize there's absolutely no music in the fight scene or most of the train scenes, and most of this movie is completely musicless.
I don't know why they didn't include any music in the fight scene, that people say is so intense (I didn't find it that interesting), but some music might have actually added a great deal to the tension and (supposed) excitement of the scene.
Some of the video games are like that too haha. Open a door, Bond music plays. Grapple hook onto a platform, Bond music plays. Activate some random switch, Bond music plays.
Yeah I noticed back then they really liked the save the James Bond theme for the really exciting sequences such as: Bond gets off a plane, Bond checks into a hotel room, Bond makes himself a sandwich.