When accepting public keys from someone setting up an identity provider for access to resources protected by a service provider using SAML 2.0, do you absolutely need to have a unique certificate? Is this covered in the SAML specifications? 

If they don't, I assume that use of certificates as a layer of defense is rendered void. An example might be someone setting up a test IdP and reusing the certificate for production.

Yes, clients are vulnerable to attack.
The initial security notices indicated that a malicious server can use the Heartbleed vulnerability to compromise an affected client. Sources below (all emphasis is mine).
Since then, proof of concept attacks have validated this position - it is utterly certain that clients running apps that use OpenSSL for TLS connections may be vulnerable.
heartbleed.com:

...When [Heartbleed] is
exploited it leads to the leak of memory contents from the server to
the client and from the client to the server.

Ubuntu Security Notice USN-2165-1:

An attacker could use this issue to obtain up to 64k of memory
contents from the client or server

RFC6520:

5. Use Cases
Each endpoint sends HeartbeatRequest messages...

OpenSSL Security Advisory 07 Apr 2014:

A missing bounds check in the handling of the TLS heartbeat extension
can be used to reveal up to 64k of memory to a connected client or
server.


Client applications reported to be vulnerable (Credit to @Lekensteyn except where otherwise stated):

MariaDB 5.5.36
wget 1.15 (leaks memory of earlier connections and own state)
curl 7.36.0
git 1.9.1 (tested clone / push, leaks not much)
nginx 1.4.7 (in proxy mode, leaks memory of previous requests)
links 2.8 (leaks contents of previous visits!)
All KDE applications using KIO (Dolphin, Konqueror).
Exim mailserver
OwnCloud Version Unknown | Source

Note that some of these programs do not use OpenSSL. For example, curl can be built with Mozilla NSS and Exim can be built with GnuTLS (as is done on Debian).

Other common clients:

Windows (all versions): Probably unaffected (uses SChannel/SSPI), but attention should be paid to the TLS implementations in individual applications. For example, Cygwin users should update their OpenSSL packages.

OSX and iOS (all versions): Probably unaffected. SANS implies it may be vulnerable by saying "OS X Mavericks has NO PATCH available", but others note that OSX 10.9 ships with OpenSSL 0.9.8y, which is not affected. Apple says: "OpenSSL libraries in OS X are deprecated, and OpenSSL has never been provided as part of iOS"

Chrome (all platforms except Android): Probably unaffected (uses NSS)

Chrome on Android: 4.1.1 may be affected (uses OpenSSL). Source. 4.1.2 should be unaffected, as it is compiled with heartbeats disabled. Source.

Mozilla products (e.g. Firefox, Thunderbird, SeaMonkey, Fennec): Probably unaffected, all use NSS


