If you've been sponsored for U.S. immigration (actually, the proper word is petitioned), your sponsor/petitioner was most likely either a U.S. employer or a family member who is a U.S. citizen or lawful permanent resident. That person or entity needs to start the process for you; and follow through to the finish.
At the early end of the process, the whole reason this person or entity is called the "petitioner" is that the first application that must be filed with U.S. Citizenship and Immigration Services (USCIS) to start off your immigration process is a petition (usually on Form I-140 if the employer is the petitioner, or I-130 if a family member is). Your family member or employer prepares and signs it, without your help.
But what happens if your petitioner has a change of mind partway through, and wishes to stop cooperating with your efforts to immigrate? The answer will depend on various factors, in particular how far along you are in the process when that happens.
Your petitioner can, at any time, inform USCIS of a intention to withdraw the petition. If the petition has not yet been approved, USCIS will almost certainly deny or cancel it, and you will not be able to go forward with your plans to immigrate.
An exception exists, however, under the Violence Against Women Act (VAWA), for battered spouses and children of U.S. citizens or permanent residents. Congress created this exception to address situations where U.S. petitioners were attempting to control the lives of their immigrant spouses and children and keep them silent about the abuse, by perpetually threatening to withdraw the I-130 petition filed on the basis of the marriage. Such immigrants may "self-petition," or file for U.S. residence on their own behalf, using USCIS Form I-360.
If you got as far as an approved petition (most likely Form I-130 or I-140), that's good, but it might not ultimately help much. You will need further cooperation from your petitioner in order to complete the green card application process.
For example, in a family-based green card application, your family member will need to promise the U.S. government to provide financial backup for you for approximately ten years, in order to avoid the possibility that you will need to receive public assistance. This is done on Form I-864, the Affidavit of Support.
In an employment-based green card application, your petitioning employer will need to provide a last-minute letter verifying that the job is still open.
Again, battered immigrant spouses and children may continue the process without their petitioner's support.
If you have already obtained U.S. residence (either through consular processing or adjustment of status), and you didn't commit fraud in the process, you probably don't need to worry. Even if your petitioner attempts to withdraw support, it will not likely affect your immigration status.
However, if your sponsor alleges and proves that your immigration petition or green card application was based on fraud, then USCIS will take action and you can be removed from the U.S. (deported).
This is possible even after you have obtained a green card. For example, if you entered into a fraudulent marriage with a U.S. citizen and the citizen decides to come clean, or if a U.S. citizen faked being your sibling, you could lose your green card. At this point, you'd need to find a legal strategy to make your case at a deportation (removal) hearing.
If instead of permanent residence you obtained conditional residence based on marriage to a U.S. citizen, the situation gets a little trickier. You and your spouse would normally be expected, near the end of your two-year period of conditional residency, to submit a joint petition on Form I-751, signed by both of you and enclosing various documents showing that you're still married.
The I-751 petition asks USCIS to grant you permanent, rather than conditional residence. The problem is that a spouse who is no longer interested in helping you immigrate will refuse to sign this petition. Fortunately, various waivers are available, including for battered spouses, immigrants who would face extreme hardship upon removal to their home countries, or spouses who entered into a bona fide marriage that was ended by divorce. See, for example, I-751 Waiver for Conditional U.S. Residents in a Divorce.
If you are facing a situation where your U.S. petitioner is threatening to withdraw the petition, refusing to sign onto a joint I-751 petition, or otherwise won't cooperate in your case, speak to an experienced U.S. immigration attorney. The attorney can fully analyze what stage of the process you are at, what your rights are at this point, and what the best strategy might be going forward.