It is often said that RSA encryption with a private key is the same as signing (signature generation). Will RSA encryption with a private key over a cryptographic hash give the same result as performing a signature generation operation?
No, RSA encryption with a private key is not the same as RSA signature generation. RSA encryption can only be performed with an RSA public key according to the RSA standard.
The terms Raw RSA or textbook RSA are often used to indicate RSA without a padding scheme. Raw RSA simply consists of modular exponentiation. Raw RSA is vulnerable to many cryptographic attacks and is not safe for general use.
It is often mentioned that signing is equivalent to RSA encrypting (the hash over) the message using the private key. This is only true if you disregard the required padding mechanism. The RSA padding mechanisms are different for encryption and signing.
A deeper look into the PKCS#1: RSA Cryptography Specifications
The older PKCS#1 v1.5 standard contains two padding schemes that are often simply referred to as PKCS#1 v1.5. The padding for these encryption and signature schemes are however quite different, which is reflected by their official names RSAES-PKCS1-v1_5 and RSASSA-PKCS1-v1_5. Fortunately that confusion is not present for the newer OAEP encryption - which uses RSAES-OAEP padding - and PSS signature generation schemes - which uses RSASSA-PSS padding.
The actual modular exponentiation is mathematically the same for RSA encryption with a public key and RSA signature generation using the private key. This is easily verified by looking at the last part of paragraph 5.2 [emphasis mine]:
In general, a signature created by performing RSA encryption will fail if the other party correctly implemented the verification method.
However, some software libraries actually perform RSA padding for signature generation if the private key is used for encryption. One possible reason for this is SSL/TLS; versions of TLS up to v1.2 used a "signature" created from an MD5 hash concatenated with a SHA-1 hash. Such a scheme is often not compatible with the signature generation methods provided by RSA implementations, hence they rely on the RSA encryption routine to magically do the right thing.
Protection against information leakage
Finally note that public keys do not require protection against information leakage, while private keys should be kept secure at all times. Hence RSA encryption implementations - when programmed to be used with a public key - may not contain protection against side channel attacks, possibly exposing the private key to an attacker that is able to apply a side channel attack.
It depends on what you mean by RSA. If you mean the plain textbook RSA where $P = C^d \bmod n$ (decryption with private key $d$) and $S = M^d \bmod n$ (signature generation), then yes, they are the same.
However, textbook RSA is inherently unsafe, and for real-life RSA such as RSA-OAEP+ (encryption) or RSA-PSS (signatures) signing is not the same as decryption.