The Sydney Funnel Web Spider's venom is called δ-atracotoxin and it is especially toxic to primates. It is a peptide toxin, which means that it has the potential to be digested in the stomach before having any physiological affect. However, many peptide toxins are not inactivated by swallowing--you don't want to be eating any Botulinum toxin.
Characterization of insecticidal peptides from venom Australian funnel-Web spiders (Vonorax et al. 2006) harvested venom from Hadronyche infensa, the Darling Downs funnel-web spider. This spider is closely related to the Sydney Funnel Web spider and also produces δ-atracotoxin.
An oral ingestion activity test was undertaken using crude H. infensa venom.
The ingestion test was performed with insect larvae, which are susceptible to the venom when injected. (Aside: there's a lot of interesting research on the toxin's effect in insects). They didn't find much:
Oral ingestion activity test of crude H. infensa venom
No effect was observed in this test, even though approximately 4 times the dose required for lethality by injection was orally consumed by the larvae.
I don't think anyone has performed a study of toxin ingestion in primates (if they have, I couldn't find it). The researchers don't speculate as to why the toxin has no effect when ingested. As expected, the toxin had no effect on (non-primate) mice:
Effect of atracotoxin-Hv1 on newborn mice
All mice in both test and control groups survived without apparent toxic effect beyond 24h post injection.
Whether this is applicable to other spider venom depends on the nature of the venom. A non-peptide, low molecular weight venom able to be absorbed by the lining of the esophagus could be very toxic by ingestion.
Scanning Mutagenesis of ω-Atracotoxin-Hv1a Reveals a Spatially Restricted Epitope That Confers Selective Activity against Insect Calcium Channels (Tedford et al. 2004)
Isolation and pharmacological characterisation of δ-atracotoxin-Hv1b, a vertebrate-selective sodium channel toxin (Szeto et al. 2000)
(for more info on peptide toxins): Cyanobacterial Peptide Toxins