Notation : given $a$, we call any number formed by inserting $\pm$ signs to the left of every number and evaluating, a "resultant" of $a$. Note that the set of resultants is symmetric i.e. $b$ is a resultant of $a$ if and only if $-b$ is (by switching signs) so we focus on just the positive resultants.
Note that if $a$ has an even number of digits, and all its digits are odd, then any resultant is even, and therefore clearly cannot divide $a$. Hence, $a$ is hopeless. One can see just from here that the number of hopeless numbers is infinite. E.g. $79,97,1531$ are all hopeless.
Furthermore, let $a$ be hopeless. Then, consider $10^ka$ for any $k \geq 1$. Note that the resultants of $10^ka$ coincide with the resultants of $a$. The factors of $10^ka$ are the original factors of $a$ multiplied by powers of $2$ and $5$ less than $k$. Suppose all resultants of $a$ are coprime to $2$ and $5$. Then, certainly none of these can be a factor of $10^ka$ as well, rendering $10^ka$ hopeless.
For example, consider a non-trivial hopeless number $74$. Its resultants are $11$ and $3$ (we can leave the negative resultants out),none of which are divisors of $74$, and both of which are coprime to $2$ and $5$. Consequently, the numbers $74\times 10^k$ gives an infinite family of hopeless numbers.
Other families of hopeless numbers can be formed by looking at large enough squares of primes, because the only divisors of such a number are the number, the prime and $1$. If somehow $1$ is not a resultant of such a number (remains to be seen when we can be sure of this) then the number is hopeless. Note that we can rephrase "$1$ is a resultant" as "there are two subsets of digits of $a$ whose difference is $1$".
To make the enough above precise, consider any number bigger than $37$. The resultants of its square cannot be more than $9$ times twice its number of digits, which if the number is $> 37$ ensures that no resultant can reach the number. Therefore, we may search in this domain. Hopeless numbers are easily found , like $41^2 = 1681,43^2 = 1849$ etc. (We cannot say if there are infinitely many of these, however).