Will moving my web site to another web host change my web site ranking position on search engines?

NO! The rankings of your web site will not change if you shift your web site to another hosting company. What matters is your domain name. As long as you keep your domain name, the rankings of your web site on different search engines will not change.
I obviously assume that the new web hosting server will be as good, if not better, as the older one.

Search engines are concerned only with the URL of the web pages on your site; if these change, your web site rank will change.

However, the location of the web server that hosts your web site might be a factor in determining rankings. On this point you would also like to read - Should I put my web site on a web hosting server located in my country?

The point I'm trying to convey is that if you do not change the URLs of the web pages on your site, the ranking will not be affected.
If the web page addresses are modified, you will lose all the advantages of in-bound links (if the pages had any). The number of in-bound links is an important factor when determining the web page rankings by search engines especially Google.

So if you are dissatisfied with your present hosting go ahead and change your web site server. For further details, read - "I'm not happy with my present web hosting. Can I change it?" and "How do I change my web hosting server?"

How can your web site rankings change if you shift your web hosting server

As I mentioned above, it does not matter to search engines if your site is hosted on a server in California or one in Antarctica as long as the URLs of your web pages remain the same. However, if you shift to a bad web server, the rankings might change. A bad web server would be one that is down or offline frequently. This is important. You should always check the uptime guarantee offered by the new web host. It should be the same as your present host (if not more). Because if search engine bots find your web pages offline most of the times, the rankings are sure to come down... right?

Also, a server on which your web site pages download slowly can be deemed as bad. In such cases, search engine bots might not be able to access and index your web site completely.

Some important points

When changing hosting or even during web site maintenance, DO NOT change the URLs of your web pages. Search engines put a lot of emphasis on in-bound links. If you modify a web page URL, you lose the advantage of in-bound links. You can come to know which pages on your web site have in-bound links through the Google Webmaster Central and especially the Google webmaster tools. You would need to open a Google account which is free.

Final point: To know what Google has to say about shifting web hosting or I.P. address, read Will switching hosting companies or changing my IP address affect my listing in Google?