This opening move does little for development or control of the center. However, this can be used as an effective waiting move, though 1. Nf3 is a lot better. Also, if White knows a lot of theory as Black, this would be useful. 1. h3 also blocks a possible Bg4 by Black. This may be quite a playable opening despite being inferior to many other openings (like 1. e4, 1. d4, etc). Because it is inferior, and most masters want to exploit their first move advantage as White, this is rarely played, except in amateur level.
1... e5 followed by 2. a3?! is the Global Variation and if 2... d5 is played after that, it transposes to the creepy-crawly formation of the Anderssen Opening (1. a3). However, the best follow-up for 1... e5 is 2. c4, transposing into the King's English Variation of the English Opening. However, 2. d4 can be tried, transposing into a "reversed" Scandinavian Defense, also playable.
1... d5 would be best followed by 2. d4, 2. f4, or 2. c4. 2. b4 would be interesting. Another interesting option is 2.g4 transposing to the Killer Grob Attack.
1... f5 is not recommended, as White can play 2. d4, transposing into the Korchnoi Attack of the Dutch Defense (another way to try to exploit kingside weaknesses created by Black, but Black has enough resources to withstand it after 3. g4 [the original followup plan for h3, especially after 3... d5, Black gets a good game, so White has to exploit the weaknesses later on in the game]). Nevertheless, it is playable as long as Black is careful.
1... c5 is an unexplored option that would lead to unique lines of the Sicilian Defense after 2. e4.
1... Nf6 is a good response, where if White plays 2. d4 or 2. c4, the game could take upon a character of the King's Indian Attack (where Black is playing the King's Indian Attack because 1. h3 wasted a tempo.)
1... Nc6 is an interesting option that could be quite playable as well.