There is not enough information in this single record to directly answer the question about your great-uncle's death or burial.
Let's look at this record a little more closely.
Ancestry's database U.S., Social Security Applications and Claims Index, 1936-2007 has the following information for where the data came from:
Original data: Social Security Applications and Claims, 1936-2007.
Looking at the URL for the record, we have another clue -- it contains the word Numident. The Social Security Administration describes the Numident records as follows:
A Numident is a query display of the information taken from an
individual's application for an original SSN card and subsequent
applications for replacement SSN cards.
Ancestry's About the database section explains the different types of records that can be found in this index-only collection:
You may also find details on changes made to the applicant's record,
including name changes or information on claims that were recorded.
The most common types of claims noted include:
- Original SSN (when the original application was submitted to obtain a SSN)
- Life Claim (when a claim was made for disability or retirement benefits)
- Death Claim (when a claim was made by a surviving family member for death or survivor benefits)
- Duplicate SSN (usually used when an application was made to replace a SS > card, it may also indicate a change in SSN or that more than one SSN was
This record is from a Life Claim, which may explain why no death date is included. Nor does it include a Social Security Number. It tells us that your great-uncle was likely to be alive on 09 Sep 1954, and applied for some kind of benefit -- probably retirement benefits, since on the claim date he would have been over 68 years old.
A quick search of the Social Security Death Index at Ancestry did not bring up any results. There are several reasons why this might happen. His record might be in the index but badly indexed, or he might not be in the index at all. The FamilySearch Research Wiki's article says:
The "Social Security Death Index" is an online searchable database. It
only includes the names of deceased individuals whose deaths were
reported to Social Security. Beginning in 2014, rules governing the
"Social Security Death Index" changed. Records for the most recent 3
years are not available.
This index is a master index file of deaths reported to the Social
Security Administration. It has been kept since 1962, when operations
were computerized. The index includes about 50 percent of deceased
persons from 1962 to 1971 and about 85 percent of deceased persons
from 1972 to 2005. It also includes a few deaths from 1937 to 1961.
If you want to request whatever information the Social Security Administration may have, you can do so by making a FOIA (Freedom of Information Act) request. Since your great-uncle was born over 120 years ago, his record does not fall under the privacy restrictions and should not be redacted, according to the information on that page. They say:
You can request a copy of the following, using online Form SSA-711.
But let's assume that you do that, and all the information they send is what Ancestry has shown you already. Now what?
Source record checklists such as Sources of Genealogical Information suggest other places besides the SSDI where you can look for information about someone's death. But the basic principle in genealogy and family history is to start with what you know, and to work from there -- and it is more productive to try and make small steps instead of big leaps.
We 'know' that he was likely to be alive on 10 Sep 1954, and from your tag, I see that you think he lived or died in New York City. Assemble all the information you have, noting how you know what you have learned so far. Then look for any information you can find around the early 1950s -- residence information, the names of other living family members, prior employers -- anything that might help you identify a record about your great-uncle.
Death records are always difficult to find because there is no guarantee that people will die in the community where they live. If they are out of state visiting a relative, the death certificate will be in the county where they died, not in their home county.
If you are searching newspapers for obituaries, the deceased person's name might not be spelled properly, or the underlying OCR might be defective. I have found many obituaries by searching for the name of the siblings or other survivors instead of the deceased. Obituaries can also be published in the towns where the survivors lived, as well as the town where the deceased lived.
Start in 1954 and see if you can inch your way forward to narrow the possible date of death.
In late 2016, the US National Archives added some information from the Social Security Administration's Numident files to their Access to Archival Databases (AAD) website.
The Scope and Content notes explain what data is in the database:
Scope & Content Note: This series contains records for every social security number (SSN) assigned to individuals with a verified death or who would have been over 110 years old by December 31, 2007. There are three type of entries in NUMIDENT: application (SS-5), claim, and death records. A NUMIDENT record may contain more than one entry. Information contained in NUMIDENT records includes: each applicant's full name, SSN, date of birth, place of birth, citizenship, sex, father's name, mother's maiden name, and race/ethnic description (optional). NUMIDENT includes information regarding any subsequent changes made to the applicant's record, including name changes and life or death claims. The death records in NUMIDENT do not include any State reported deaths in accordance with the Social Security Act section 205(r). There are 72,182,729 SS-5 records entries; 25,230,486 claim record entries; and 49,459,293 death record entries.
These files come from the same source as Ancestry's U.S., Social Security Applications and Claims Index, 1936-2007, but the data is presented in a different way, and in some cases, reveal more information than Ancestry does. However, not all deaths are in the SSA's Death Master File, so your great uncle's death date may not be in these files.
The instructions on ordering NYC death certificates and the restrictions on who can receive the "long form" death certificates with cause-of-death information or only the short form are on this page at the NYC Department of Health: Death Certificates
If you have difficulty finding the exact record you seek, look for other records that can supply the same information, using a checklist or record finder such as:
As you consult general research guides like these, keep in mind that New York City has different record-keeping than the rest of New York State, so advice about finding records for New York State often won't apply to New York City.
You could also consider hiring a professional genealogist who is experienced in the area. See this answer for links and guidelines.