I was trying to calculate RSI over hourly OHLC bars for a symbol (AAPL as an example) and got stock, first how should I handle closing hours? (does it make sense to ignore them all together and assume that next day's data is following today's data [2018-01-01/16:00 -> 2018-01-02/09:00 -> 2018-01-02/10:00]) and second how should I handle holidays and weekends? (assuming I'm going to use daily data instead of the hourly data)
Traditionally, you should have one continuous time-series for any frequency of data and calculate your indicator based on that. That is, unless you have some other desired or proprietary method for calculating your indicators.
You should generally exclude times when the market is closed, such as weekends and holidays, as this would produce undesired bias to the indicator's value.
So in your case, calculate the RSI based on a single continuous time-series that does not include weekends, holidays or times when the market is not open.
Note - as you probably suspected, gaps in the overnight price can have a significant impact on the reading of your indicator, especially depending on the frequency of the data. So, if you are making trading decisions based on this information, you should keep that in mind or have some sort of rules based criteria in the event of significant overnight changes.