As selfish as it may seem to you, there are any number of good reasons why an individual may be a bit wary of sharing their treasured family heirlooms or hard-earned genealogy research. The trick to encouraging these people to share is understanding their possible reservations and treating them in a polite and respectful manner.
Offer something in return. This may seem like just common sense, but the trick sometimes is to figure out what you can offer that will be useful. If you'd like to see your great-grandmother's Bible, offer to bring over a computer and scanner and scan the Bible in person. Your relative can even do all of the physical handling of the Bible if that makes her happy. Then create a CD with the digitized pages she can share with any other relatives who ask. This technique works well for family photographs and records as well. When requesting information from a fellow researcher, ask if there is anything on your family they might be interested in, or offer to do lookups for them in any related records or resources you own or have access to. Even if you have nothing unique to offer that particular researcher, you can still return the favor by helping others on mailing lists covering your mutual surname or geographical area of interest.
Give credit where credit is due. It may be frustrating to find someone who won't just send you a copy of their entire family file. Why not? It's your family tree too! It is quite understandable, however, when you consider that many genealogists have graciously shared their hard-earned research with a curious "cousin" only to find it plastered all over the Internet without credit or acknowledgement. For many of us, our personal research is the product of hundreds, even thousands, of hours -- not to mention the monetary cost -- and it can make us a bit cranky to see it used in such a cavalier manner. Anytime you receive information from someone else, be sure to note them as the source of the information.
Do your homework before asking for help. It's amazing how many emails I receive from people researching the POWELL surname wanting to know if I have any information on their family. It may be my last name (and my married name at that), but do you realize how many POWELLS have lived in the world? You're not going to get very far on your family tree by expecting everyone else to do the work for you. Trace your own line back as far as you can, and make sure that any emails you send out or queries you post are at least related to your surname in the correct geographic area and time period. Include as many details as possible so that others can tell at a glance how your line might fit in with theirs.
Make it personal. As much as genealogists love everything having to do with family history, they are most interested in their own names, not yours. Instead of sending out a general request for information on your great-grandfather or your POWELL surname, consider wording your query in terms of how your great-grandfather or POWELLS might be related to their family line. It's not that other genealogists won't provide you with information on families or research that doesn't relate to theirs - genealogists, as a group, tend to be extremely nice and helpful. But genealogists also get busy, and emails or posts with subject lines that offer a potential link to their own research stand a greater chance of catching their eye.
Respect the privacy of living relatives. One of the quickest ways to alienate your relatives is to share or publish the personal information they have entrusted to your care without their knowledge and consent. It's fine to include it in your working genealogy file, but DO NOT publish it on the Internet or in a book, or otherwise share it without their permission. Older relatives, especially, are a bit overwhelmed by the Internet and its threat of identity theft. Before sharing a GEDCOM with a third-cousin or posting it to an online database, use the feature in your genealogy software program that hides information about living people, or use a GEDCOM filter to exclude all people born within the past 100 years. It can also be helpful to create a special field in your genealogy software program for information which living relatives have requested you keep private -- perhaps a story of a family feud or a child born out of wedlock -- and then exclude that field when creating GEDCOM files for sharing with others. If you want your relatives to feel comfortable sharing personal information with you, then you need to show they can trust you to keep it confidential when necessary.