Short Answer(as it applies to humans):
Not in short exposures.
Polyvinyl acetate (PVA or PVAc) is the substance that makes up the active portion of Elmer's / school / wood glue -- the rest being water. Sodium borate (borax) can be used to cross-link the polymers present in the emulsion to form a slimy substance.
Some polymer cross-linking processes can be exothermic, but I'm going to set that aside for now because... extended exposure to borax alone can cause the symptoms described. Some of the articles relating to the incident linked to in the question state that the child had been making and handling this material for weeks.
There are a few studies in which volunteers were given lotion/shampoo containing >1% sodium borate. After about a week, some experienced the described symptoms.
A cumulative irritancy test was used to study the effect of repeated exposures
of a hair preparation containing 3.2% Sodium Borate and a cleansing cream containing
1.7% Sodium Borate on the skin of 12 and 14 subjects, respectively (Table
2). The test material was applied under an occlusive patch to the backs of subjects
daily for 21 consecutive days. Sites were scored one hour after patch
removal. Applications 4 to 21 of the hair preparation produced erythema and
papules in most subjects; the total cumulative irritancy score was 571 (maximum
score = 630). The cleansing cream caused slight erythema in two subjects only,
resulting in a total irritancy score of 6.4. The investigators concluded that, under
the conditions of the study, the hair preparation was a "mild to moderate"
cumulative irritant, whereas the cleansing cream was practically non irritating.
In a similar study, the contact-sensitizing potential of a cleansing cream containing
1.7% Sodium Borate was tested in 22 subjects (Table 2). Preliminary irritancy
testing revealed no irritation to a 48 h patch containing this product;
therefore, skin sites were pretreated with 5% SLS for 24 h prior to application of
the initial induction patch. The product was applied under occlusion for 48 h,
every other day for 10 days (five applications). Following a 10- to 14-day rest,
48-hour occlusive patches containing the test material were applied to fresh sites
(with and without SLS pretreatment). SLS controls were also applied. Sites were
scored at 48 and 72 h. More than half of the subjects reacted to SLS treatment
(with or without application of test material). No significant irritation was observed
at sites tested with the cleansing cream alone. The product containing 1.7%
Sodium Borate was nonirritating and nonsensitizing.
In summary, this material should be pretty safe for short term handling with unbroken skin. Continuous use increases the risk of mishandling (broken/compromised skin contact) leading to boron absorption which may result in skin irritation (redness, bumps).
Note: Abstract does not mention the specific study referenced.