Are you sure that the strings in the vector you lapply over are in the column names of the icu dataset? 
It works for me when I download the icu data:
system("wget http://course1.winona.edu/bdeppa/Biostatistics/Data%20Sets/ICU.TXT")
icu <- read.table('ICU.TXT', header=TRUE)
and change status to STA which is a column in icu. Here an example for some of your variables:
my.list <- lapply(c("Age","Sex","Race","Ser","Can"),         
       function(var) {
         formula    <- as.formula(paste("STA ~", var))
         res.logist <- glm(formula, data = icu, family = binomial)
         summary(res.logist)
       })
This gives me a list with summary.glm objects. Example:
lapply(my.list, coefficients)
[[1]]
               Estimate Std. Error   z value     Pr(>|z|)
(Intercept) -3.05851323 0.69608124 -4.393903 1.113337e-05
Age          0.02754261 0.01056416  2.607174 9.129303e-03
[[2]]
              Estimate Std. Error    z value     Pr(>|z|)
(Intercept) -1.4271164  0.2273030 -6.2784758 3.419081e-10
Sex          0.1053605  0.3617088  0.2912855 7.708330e-01
[[3]]
              Estimate Std. Error    z value   Pr(>|z|)
(Intercept) -1.0500583  0.4983146 -2.1072198 0.03509853
Race        -0.2913384  0.4108026 -0.7091933 0.47820450
[[4]]
              Estimate Std. Error   z value     Pr(>|z|)
(Intercept) -0.9465961  0.2310559 -4.096827 0.0000418852
Ser         -0.9469461  0.3681954 -2.571858 0.0101154495
[[5]]
                 Estimate Std. Error       z value     Pr(>|z|)
(Intercept) -1.386294e+00  0.1863390 -7.439638e+00 1.009615e-13
Can          7.523358e-16  0.5892555  1.276756e-15 1.000000e+00
If you want to do a chi-square test:
my.list <- lapply(c("Age","Sex","Race","Ser","Can"),function(var)chisq.test(icu$STA, icu[,var]))
or a chi-square test for all combinations of variables:
my.list.all <- apply(combn(colnames(icu), 2), 2, function(x)chisq.test(icu[,x[1]], icu[,x[2]]))
Does this work?