Does signing multiple messages with the same RSA key weaken the encryption?
I'll ignore the term "encryption" in the title, since it is improper: signature is not encryption. Instead I'll try to answer the question:


  Does direct textbook RSA signing of hashes of 1000 messages $M_j$ per the textbook RSA signaing function $\mathcal S$, as $S_j=\mathcal S(H(M_j))=H(M_j)^d\bmod N$, weaken the resistance against signature forgery for other messages under the same scheme, given $(N,e)$ a public key with 1024-bit RSA modulus.


Yes, for too narrow a hash $H$. One attack strategy could be that of Yvo Desmedt and Andrew M. Odlyzko: A chosen text attack on the RSA cryptosystem and some discrete logarithm schemes, in proceedings of Crypto 1985.

The basic attack


Factors the $H(M_j)$ and keeps those that are $B$-smooth, forming a (sparse) matrix $A$ with one column per such $j$, one line for each prime $p_i$ that appears in such factorizations, collectively forming the factor base. The multiplicity of prime $p_i$ in $H(M_j)$ is at line $i$ column $j$ of $A$. 
Factors many $H(M'_k)$, and for those which factors all are among the factor base, form the vector $V$ of the multiplicities of that $H(M'_k)$, and try to solve for $X$ the system $V=A\cdot X$ where $X$ is a vector with one entry per column in $A$, and operations are in $\Bbb Z_e$.
Once such $M'_k$ is found, computes the signature of $M'_k$ using


the multiplicative property: $\mathcal S(H\cdot H'\bmod N)=\mathcal S(H)\cdot\mathcal S(H')\bmod N$,
the known signatures $S_j$,
the coefficients in $X$, corresponding to powers of the $S_j$,
the fact that forall $p_i$, it holds $\mathcal S({p_i}^e\bmod N)=p_i\bmod N$.



It can work, or not, depending on:


The width of the hash. The narrower, the more smooths, the easier the attack.
The value of $e$. Lower $e$ (e.g. $e=3$) make the attack easier.
If the adversaries are able to choose some of the $M_i$; this allows the adversaries to try much more than 1000 messages in step 1, somewhat offsetting the effect of a wider hash.


In an improvement extending the applicability to less signed messages or wider hashes, pairs of $H(M_j)$ having a single prime factor larger than $B$ are kept separately if another $H(M_{j'})$ is in this case and shares the same larger prime factor, with the difference modulo $e$ in the respective multiplicities of the primes in the factor base for $H(M_j)$ and $H(M_{j'})$ used for an additional column in $A$. A further improvement uses two large primes.

It can be predicted about how many messages are required for a given hash width and $e$, and the work necessary. I wish I knew exactly how! There is an analysis in appendix C of the full version of Jean-Sebastien Coron, David Naccache, Mehdi Tibouchi and Ralf-Philipp Weinmann's Practical Cryptanalysis of ISO/IEC 9796-2 and EMV Signatures, in proceedings of Crypto 2009, but it does not account for $e$, nor of the large prime(s) improvement, nor of working under the constraint of relatively few non-chosen signed messages. 


  would a padding scheme protect against it?


Yes, by making the input of the textbook RSA signature function $\mathcal S$ too large to allow the attack.