Will Oracle ignore my index?
Depends on the data distribution. I can easily construct examples for both cases (index vs table scan).

Just think about the case when the columns are unique or nearly unique (index), or when all the rows are the same (table scan).

Technically your index can be used for the above query.

Given the general nature of a column named ID, I would say the index will be chosen, but it is not guaraanteed.

The predicate DATE_TIME is not null will be processed at the table level, not the index level, even if you make sure that the NULL rows are indexed as well:

create index index_1 on table_1(id, name, trunc(date_time), 1);

PLAN_TABLE_OUTPUT
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Plan hash value: 852675278

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| Id  | Operation                           | Name    |
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|   0 | SELECT STATEMENT                    |         |
|*  1 |  TABLE ACCESS BY INDEX ROWID BATCHED| TABLE_1 |
|*  2 |   INDEX RANGE SCAN                  | INDEX_1 |
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Predicate Information (identified by operation id):
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   1 - filter("DATE_TIME" IS NOT NULL)
   2 - access("ID"=1 AND "NAME"='Oracle')