I have seen this posted across many forums and I am sure many of you would be curious to know the answer to this question too. I believe that Web Hosting does not directly affect your websites search engine performance but indirectly it does.
See it does not really matter which company is hosting your website or the platform your website is hosted on. It can be Linux, Windows or whatever operating system you choose your website to be on, it does not matter. But there are a few obvious factors that will affect your search engine optimization efforts. I am listing a few as they come to my mind below:
If you have a website hosted on a very slow and sluggish web server then you don’t need to be a genius to know that you website will server slower to your end users compared to a website hosted on a fast, high performance server. Search engines do keep a note of the time it takes for your site to open each time a browser makes a request for your website. It will register your site as a heavy site. Further, more customers will leave your site before it even opens. This is something which is also noted by search engines. They make a note of how long a customer stays on your site. If more people leave your site too frequently that means your site has nothing noteworthy to offer to your customers and it will be assumed that you do not have good content to keep your customers on your website. Definitely a negative point in terms of optimization.
Black Listed Servers:
If your site is hosted on an IP that sends out spam or hosts illegal or adult websites, your website too will be seen in the same light. Which is definitely not good.
Servers that are frequently down:
There are hosting providers around with servers that has less time and more downtime. Stay away from such service providers. A website not found or down is equally bad for search engines. Imagine Google trying to crawl / index your site and it finds that your site is not available at all.
These are a few points that can affect your website search engine visibility or popularity immensely.