Using OpenLayers doesn't depend in any way on the hosting machine.
So I assume you would like to set up a geographic server.
The things you should care about are:
- processing power (CPU), which is needed to create map tiles and/or query data
- RAM, which allows many processes to work in parallel and is linked to CPU
- disk storage is important if you intend to have your server store a large number of tiles (a WMS server)
- bandwidth, necessary to have your tiles served fast
I never used ArcGIS Server nor QGIS Server (I'm using GeoServer) but the above mentioned aspects should anyway be considered.
I would suggest renting a hourly based low performance virtual machine for testing and then eventually upgrade it based on the outcome.
Something I didn't mention before but I think is relevant to the whole question is: "How many concurrent users is the app going to serve?".